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New solutions to local and non-local elliptic equations Chan, Hon To Hardy 2018

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New solutions to local and non-local elliptic equationsbyHon To Hardy ChanB. Sc., The Chinese University of Hong Kong, 2010M. Phil., The Chinese University of Hong Kong, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)June 2018c© Hon To Hardy Chan, 2018 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  New solutions to local and non-local elliptic equations  submitted by Hon To Hardy Chan  in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics  Examining Committee: Juncheng Wei, Mathematics Co-supervisor Nassif Ghoussoub, Mathematics Co-supervisor   Supervisory Committee Member Ailana Fraser, Mathematics University Examiner Gordon Semenoff, Physics University Examiner   Additional Supervisory Committee Members: Jingyi Chen, Mathematics Supervisory Committee Member Michael Ward, Mathematics Supervisory Committee Member  AbstractWe obtain a few existence results for elliptic equations.We develop in Chapter 2 a new infinite dimensional gluing scheme for frac-tional elliptic equations in the mildly non-local setting. Here it is applied to thecatenoid. As a consequence of this method, a counter-example to a fractional ana-logue of De Giorgi conjecture can be obtained [51].Then, in Chapter 3, we construct singular solutions to the fractional Yamabeproblem using conformal geometry. Fractional order ordinary differential equa-tions are studied.Finally, in Chapter 4, we obtain the existence to a suitably perturbed doubly-critical Hardy–Schro¨dinger equation in a bounded domain in the hyperbolic space.iiiLay SummaryJointly with my collaborators we prove that certain equations that involve calculusdo have solutions. We use two methods in finding the solutions — by looking atenergy levels, or by gluing pieces together.ivPrefaceAll the materials are adapted from the author’s research articles [7] (joint workwith Weiwei Ao, Azahara DelaTorre, Marco A. Fontelos, Maria del Mar Gonza´lezand Juncheng Wei), [49] (joint work with Yong Liu and Juncheng Wei) and [48](joint work with Nassif Ghoussoub, Saikat Mazumdar, Shaya Shakerian andLuiz Fernando de Oliveira Faria). These works are put on arXiv (respectivelyarXiv:1802.07973, arXiv:1711.03215 and arXiv:1710.01271) and are ready forsubmission. They are under review and have not yet been accepted by any journal.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Fractional Gluing on the Catenoid . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 The Allen–Cahn equation . . . . . . . . . . . . . . . . . 52.1.2 The fractional case and non-local minimal surfaces . . . . 62.1.3 A brief description . . . . . . . . . . . . . . . . . . . . . 92.2 Outline of the construction . . . . . . . . . . . . . . . . . . . . . 112.2.1 Notations and the approximate solution . . . . . . . . . . 112.2.2 The error . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 The gluing reduction . . . . . . . . . . . . . . . . . . . . 142.2.4 Projection of error and the reduced equation . . . . . . . . 192.3 Computation of the error: Fermi coordinates expansion . . . . . . 212.4 Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1 Non-degeneracy of one-dimensional solution . . . . . . . 34vi2.4.2 A priori estimates . . . . . . . . . . . . . . . . . . . . . . 382.4.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 592.4.4 The positive operator . . . . . . . . . . . . . . . . . . . . 612.5 Fractional gluing system . . . . . . . . . . . . . . . . . . . . . . 622.5.1 Preliminary estimates . . . . . . . . . . . . . . . . . . . . 622.5.2 The outer problem: Proof of Proposition 2.2.2 . . . . . . . 682.5.3 The inner problem: Proof of Proposition 2.2.3 . . . . . . . 682.6 The reduced equation . . . . . . . . . . . . . . . . . . . . . . . . 742.6.1 Form of the equation: Proof of Proposition 2.2.4 . . . . . 742.6.2 Initial approximation . . . . . . . . . . . . . . . . . . . . 762.6.3 The linearization . . . . . . . . . . . . . . . . . . . . . . 832.6.4 The perturbation argument: Proof of Proposition 2.2.5 . . 883 Fractional Yamabe Problem . . . . . . . . . . . . . . . . . . . . . . 903.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.2 The fast decaying solution . . . . . . . . . . . . . . . . . . . . . 983.2.1 Useful inequalities . . . . . . . . . . . . . . . . . . . . . 1003.2.2 Proof of Proposition 3.2.2 . . . . . . . . . . . . . . . . . 1013.2.3 Existence of a fast-decay singular solution . . . . . . . . . 1053.3 The conformal fractional Laplacian in the presence of k-dimensional singularities . . . . . . . . . . . . . . . . . . . . . . 1103.3.1 A quick review on the conformal fractional Laplacian . . . 1103.3.2 An isolated singularity . . . . . . . . . . . . . . . . . . . 1133.3.3 The full symbol . . . . . . . . . . . . . . . . . . . . . . . 1173.3.4 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 1223.4 New ODE methods for non-local equations . . . . . . . . . . . . 1293.4.1 The kernel . . . . . . . . . . . . . . . . . . . . . . . . . 1303.4.2 The Hamiltonian along trajectories . . . . . . . . . . . . . 1343.5 The approximate solution . . . . . . . . . . . . . . . . . . . . . . 1373.5.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . 1373.5.2 Approximate solution with isolated singularities . . . . . 1383.5.3 Approximate solution in general case . . . . . . . . . . . 1413.6 Hardy type operators with fractional Laplacian . . . . . . . . . . 149vii3.6.1 Beyond the stability regime . . . . . . . . . . . . . . . . 1593.6.2 A-priori estimates in weighted Sobolev spaces . . . . . . 1613.6.3 An application to a non-local ODE . . . . . . . . . . . . . 1633.6.4 Technical results . . . . . . . . . . . . . . . . . . . . . . 1643.7 Linear theory - injectivity . . . . . . . . . . . . . . . . . . . . . . 1703.7.1 Indicial roots . . . . . . . . . . . . . . . . . . . . . . . . 1723.7.2 Injectivity ofL1 in the weighted space C2γ+αµ,ν1 . . . . . . 1773.7.3 Injectivity of L1 on C 2γ+αµ,ν1 . . . . . . . . . . . . . . . . . 1803.7.4 A priori estimates . . . . . . . . . . . . . . . . . . . . . . 1803.8 Fredholm properties - surjectivity . . . . . . . . . . . . . . . . . . 1853.8.1 Fredholm properties . . . . . . . . . . . . . . . . . . . . 1863.8.2 Uniform estimates . . . . . . . . . . . . . . . . . . . . . 1953.9 Conclusion of the proof . . . . . . . . . . . . . . . . . . . . . . . 1983.9.1 Solution with isolated singularities (RN \{q1, . . . ,qK}) . . 1993.9.2 The general case Rn \Σ, Σ a sub-manifold of dimension k 2033.10 Some known results on special functions . . . . . . . . . . . . . . 2033.11 A review of the Fourier-Helgason transform on Hyperbolic space . 2054 Extremals for Hyperbolic Hardy–Schro¨dinger Operators . . . . . . 2094.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.2 Hardy–Sobolev type inequalities in hyperbolic space . . . . . . . 2174.3 The explicit solutions for Hardy–Sobolev equations on Bn . . . . 2204.4 The corresponding perturbed Hardy–Schro¨dinger operator on Eu-clidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.5 Existence results for compact submanifolds of Bn . . . . . . . . . 237Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242viiiAcknowledgmentsI express my sincere gratitude to my supervisors Prof. Nassif Ghoussoub and Prof.Juncheng Wei. I thank Prof. Juncheng Wei for his patient guidance all over theyears into this subject while I was still struggling with basic stuff, as well as comingup with numerous innovating ways and sayings to keep me working hard and fast.He included me in my first project [14] even when I was not ready to contributemuch. He also witnessed how I was humbled when I had just a little pride on myspeed totaling the midterms. I thank Prof. Nassif Ghoussoub, on the other hand,for continually caring about my progress and also pointing out the importance ofwriting papers carefully, hence slowly, when I did not to treat this seriously andmade a major mistake in a previous version of Chapter 4. I do not have enoughwords to tell the kindness I have received through them.I greatly appreciate all the examiners as well as the supervisory committeemembers, who have helped me going through the checkpoints, for all their effortsand time.I am so grateful that Weiwei Ao and Guoyuan Chen spotted some typos in thesecond draft of the thesis.It is such a blessing to have my wife and my family for their constant supportand care, without which I would not be able to complete this dissertation and otherresearch papers.It has been awesome to meet all my collaborators and friends. A full list wouldbe too long, yet among us these wonderful names should be never forgotten in thecourse of my life: Joshua Christos, Hellen Fisher, Peter Corner, Patricia Grace,Joel Nissy, Michael Kadosˇ and Amo Zechar. Let them be always remembered.For all these, thanks Living God.ixChapter 1IntroductionThe study of elliptic partial differential equations arises in many subjects includ-ing notably physics, geometry, biology and applied modeling. Solutions can beconsidered as the steady-states in reaction-diffusion systems. From the mathemat-ical point of view, the fundamental issues are the existence, regularity, uniqueness,symmetry, and other qualitative properties. In the subsequent chapters we will bedealing with non-local versions of• the Allen–Cahn equation−∆u = u−u3in phase transitions;• the Lane–Emden equation−∆u = upin astrophysics;and a local but geometric version of• the Schro¨dinger equation−∆u+Vu = upin quantum mechanics.1Non-local equations have attracted a great deal of interest in the last decade.A non-local diffusion term, usually as a model given by a fractional Laplacian,accounts for long range interactions. Intrinsic difficulties arise from the fact thatthe fractional Laplacian is in fact an integro-differential operator. They are in manycases, but not always, overcome by the Caffarelli–Silvestre extension [43], an equi-valent local problem in a space with one extra dimension, where classical tech-niques may be applied. Since then a huge amount of effort has been made in thestudy of fractional order equations.Posed by E. De Giorgi [64] in 1979, the conjecture that all bounded entiresolutions of the Allen–Cahn equation are one-dimensional at least in dimensionsn ≤ 8, has been almost completely settled: by Ghoussoub–Gui [103] for n = 2,Ambrosio–Cabre´ [12] for n = 3, Savin [154] for 4 ≤ n ≤ 8 under a mild limit as-sumption, and del Pino–Kowalczyk–Wei [67] who constructed a counter-examplefor n≥ 9.Its fractional analogue for s ∈ [12 ,1) (having taken into consideration the Γ-convergence result [157]), namely the one-dimensional symmetry of bounded solu-tions of(−∆)su = u−u3 in Rn,has also received considerable attention in low dimensions. Positive results havebeen obtained in low dimensions by Sire–Valdinoci [165] and Cabre´–Sire [37] forn= 2 and s ∈ (0,1), Cabre´–Cinti [29, 30] for n= 3 and s ∈ [12 ,1), Savin [155, 156]for 4 ≤ n ≤ 8 and s ∈ [12 ,1) again under a limit assumption, and recently Figalli–Serra [94] for n = 4 and s = 12 .In order to give a counter-example in high dimensions n ≥ 9, in Chapter 2 wedevelop a new infinite dimensional gluing method for fractional elliptic equations.As a model problem, we construct a solution of the fractional Allen–Cahn equationvanishing on a rotationally symmetric surface which resembles a catenoid and hassub-linear growth at infinity. The crux of the analysis is the fine expansion of thefractional Laplacian in Fermi coordinates and the splitting of the inner problem.Via the argument of Jerison–Monneau [120], this leads to counter-examples to DeGiorgi Conjecture for the fractional Allen–Cahn equation [51], a work that is inprogress.2The Yamabe problem asks to find a constant curvature metric in a given con-formal class [180]. This was proved by Trudinger [171] (who also discovered acritical error in Yamabe’s proof), Aubin [16] and Schoen [162]. The fractionalYamabe problem, in which a constant fractional curvature is prescribed, takes theform(−∆)su = u n+2sn−2s in Rn.We consider in Chapter 3 the problem of constructing solutions that are singular ata given smooth sub-manifold, for which we establish the classical gluing methodof Mazzeo and Pacard [132] for the scalar curvature in the fractional setting.From the way infinite dimensional gluing methods were developed, their localnature is apparent – the tangential and normal variables on the hypersurface are sep-arated. While similar technical estimates are needed in the localization by cut-offfunctions, it is essential to analyze the model linearized operator, where conformalgeometry and non-Euclidean harmonic analysis are used. Moreover, the existenceof a radial fast-decaying solution needs to be established by a blow-up argumenttogether with a bifurcation method.The Hardy–Schro¨dinger operator, whether local or non-local, has a potentialthat is homogeneous to the Laplacian. Such operator is already seen as the lin-earization of the singular solution in the fractional Yamabe problem in Chapter 3,where the infinitely many complex indicial roots are computed. In fact, variationalproblems involving such operator have their own interests.We study in Chapter 4 the existence of extremals of a non-linear elliptic Hardy–Schro¨dinger equation in the hyperbolic space. The loss of compactness due to thescaling invariance gives rise to interesting concentration phenomena. Inspired bythe recent analysis of Ghoussoub–Robert [105, 106], we obtain sufficient condi-tions for the attainability of the best constant of Hardy–Sobolev inequalities interms of the linear perturbation or the mass of the domain.The essential observation in this work is that, in the radial setting, solutionsof the hyperbolic Hardy–Sobolev equation are classified explicitly in terms of thefundamental solutions of the Laplace–Beltrami operator. With this it remains togeneralize [105, 106] to include singular perturbations.3To conclude, a strong connection between the fields of partial differential equa-tions and geometry is seen from the geometric quantities that come into play in allthe above problems. These results point to similar problems in more general set-tings, or even parabolic equations. Moreover, the gluing method devised opens upa new area of constructing solutions for non-local equations.4Chapter 2Fractional Gluing on theCatenoid2.1 Introduction2.1.1 The Allen–Cahn equationIn this chapter we are concerned with the fractional Allen–Cahn equation, whichtakes the form(−∆)su+ f (u) = 0 in Rn (2.1)where f (u) = u3− u = W ′(u) is a typical example that W (u) =(1−u22)2is a bi-stable, balanced double-well potential.In the classical case when s = 1, such equation arises in the phase transitionphenomenon [11, 45]. Let us consider, in a bounded domain Ω, a rescaled form ofthe equation (2.1),−ε2∆uε + f (uε) = 0 in Ω.This is the Euler–Lagrange equation of the energy functionalJε(u) =∫Ω(ε2|∇u|2+ 1εW (u))dx.5The constant solutions u = ±1 corresponds to the stable phases. For any subsetS ∈Ω, we see that the discontinuous function uS = χS−χΩ\S minimize the poten-tial energy, the second term in Jε(u). The gradient term, or the kinetic energy, isinserted to penalize unnecessary forming of the interface ∂S.Using Γ-convergence, Modica [140] proved that any family of minimizers (uε)of Jε with uniformly bounded energy has to converge to some uS in certain sense,where ∂S has minimal perimeter. Caffarelli and Co´rdoba [39] proved that the levelsets {uε = λ} in fact converge locally uniformly to the interface.Observing that the scaling vε(x) = uε(εx) solves−∆vε + f (vε) = 0 in ε−1Ω,which formally tends as ε → 0 to (2.1), the intuition is that vε(x) should resemblethe one-dimensional solution w˜(z) = tanh z√2where z is the normal coordinate onthe interface M, an asymptotically flat minimal surface. Indeed, we have thatJε(vε)≈ Area(M)∫R(12w˜′(z)2+W (w˜(z)))dz.Thus a classification of solutions of (2.1) was conjectured by E. De Giorgi [64].Conjecture 2.1.1. Let s = 1. At least for n ≤ 8, all bounded solutions to (2.1)monotone in one direction must be one-dimensional, i.e. u(x) = w(x1) up to trans-lation and rotation.It has been proven for n = 2 by Ghoussoub and Gui [103], n = 3 by Ambrosioand Cabre´ [12], and for 4≤ n≤ 8 under an extra mild assumption by Savin [154].In higher dimensions n ≥ 9, a counter-example has been constructed by del Pino,Kowalczyk and Wei [67]. See also [35, 104, 120].2.1.2 The fractional case and non-local minimal surfacesWhile Conjecture 2.1.1 is almost completely settled, a recent and intense interestarises in the study of the fractional non-local equations. A typical non-local dif-fusion term is the fractional Laplacian (−∆)s, s ∈ (0,1), which is defined as a6pseudo-differential operator with symbol |ξ |2s, or equivalently by a singular integ-ral formula(−∆)su(x0) =Cn,sP.V.∫Rnu(x0)−u(x)|x0− x|n+2sdx, Cn,s =22ssΓ(n+2s2)Γ(1− s)pi n2 ,for locally C2 functions with at most mild growth at infinity. Caffarelli and Sil-vestre [43] formulated a local extension problem where the fractional Laplacian isrealized as a Dirichlet-to-Neumann map. This extension theorem was generalizedby Chang and Gonza´lez [52] in the setting of conformal geometry. Expositions tothe fractional Laplacian can be found in [2, 28, 73, 109].In a parallel line of thought, Γ-convergence results have been obtained by Am-brosio, De Philippis and Martinazzi [13], Gonza´lez [108], and Savin and Valdinoci[157]. The latter authors also proved the uniform convergence of level sets [160].Owing to the varying strength of the non-locality, the energyJε(u) = ε2s ‖u‖Hs(Ω)+∫ΩW (u)dxΓ-converges (under a suitable rescaling) to the classical perimeter functional whens ∈ [12 ,1), and to a non-local perimeter when s ∈ (0, 12).A singularly perturbed version of (2.1) was studied by Millot and Sire [138]for the critical parameter s = 12 , and also by these two authors and Wang [139] inthe case s ∈ (0, 12).In the highly non-local case s ∈ (0, 12), the corresponding non-local minimalsurface was first studied by Caffarelli, Roquejoffre and Savin [41].Concerning regularity, Savin and Valdinoci [159] proved that any non-localminimal surface is locally C1,α except for a singular set of Hausdorff dimensionn−3. Caffarelli and Valdinoci [44] showed that in the asymptotic case s→ (1/2)−,in accordance to the classical minimal surface theory, any s-minimal cone is a hy-perplane for n ≤ 7 and any s-minimal surface is locally a C1,α graph except for asingular set of codimension at least 8. Recently Cabre´, Cinti and Serra [31] classi-fied stable s-minimal cones in R3 when s is close to (1/2)−. Barrios, Figalli andValdinoci [17] improved the regularity of C1,α s-minimal surfaces to C∞. Graphical7properties and boundary stickiness behaviors were investigated by Dipierro, Savinand Valdinoci [79, 80].Non-trivial examples of such non-local minimal surface were constructed byDa´vila, del Pino and Wei [63] at the limit s→ (1/2)−, as an analog to the catenoid.Note that the non-local catenoid they constructed is eventually linear, as opposedto logarithmic, at infinity; a similar effect is seen in the construction in the presentchapter.Strong interests are also seen in a fractional version of De Giorgi Conjecture.Conjecture 2.1.2. Bounded monotone entire solutions to (2.1) must be one dimen-sional, at least for dimensions n≤ 8.In the rest of this chapter we will focus on the mildly non-local regime s ∈[12 ,1). Positive results have been obtained: n = 2 by Sire and Valdinoci [165]and by Cabre´ and Sire [37], n = 3 by Cabre´ and Cinti [30] (see also Cabre´ andSola`-Morales [38]), n = 4 and s = 12 by Figalli and Serra [94], and the remainingcases for n ≤ 8 by Savin [155] under an additional mild assumption. A naturalquestion is whether or not Savin’s result is optimal. In a forthcoming paper [51],we will construct global minimizers in dimension 8 and give counter-examples toConjecture 2.1.2 for n≥ 9 and s ∈ (12 ,1).Some work related to Conjecture 2.1.2 involving more general operators in-cludes [27, 34, 81, 90, 158]. For similar results in elliptic systems, the readers arereferred to [20, 21, 74, 87–89, 91, 174, 175] for the local case, and [25, 77, 92, 176]under the fractional setting.The construction of solution by gluing for non-local equations is a relativelynew subject. Du, Gui, Sire and Wei [82] proved the existence of multi-layeredsolutions of (2.1) when n = 1. Other work involves the fractional Schro¨dingerequation [54, 62], the fractional Yamabe problem [15] and non-local Delaunaysurfaces [58].For general existence theorems for non-local equations, the readers may con-sult, among others, [53, 55, 95, 96, 116, 141, 143, 145, 146, 167, 170, 177, 178]as well as the references therein. Related questions on the fractional Allen–Cahn8equations, non-local isoperimetric problems and non-local free boundary problemsare also widely studied in [24, 42, 69, 70, 72, 75, 78, 93, 125, 127]. See also theexpository articles [1, 100, 172].Despite similar appearance, (2.1) for s ∈ (0,1) is different from that for s = 1in a number of striking ways. Firstly, the non-local nature disallows the use oflocal Fermi coordinates. Secondly, the one-dimensional solution w(z) only has analgebraic decay of order 2s at infinity, in contrast to the exponential decay whens = 1. Thirdly, the fractional Laplacian is a strongly coupled operator and henceit is impossible to “integrate by parts” in lower dimensions. Finally the inner-outer gluing using cut-off functions no longer work due to the nonlocality of thefractional operator.The purpose of this chapter is to establish a new gluing approach for fractionalelliptic equations for constructing solutions with a layer over higher-dimensionalsub-manifolds. In particular, in the second part [51] we will apply it to partiallyanswer Conjecture 2.1.2. To overcome the aforementioned difficulties, the maintool is an expansion of the fractional Laplacian in the Fermi coordinates, a refine-ment of the computations already seen in [50], supplemented by technical integralcalculations. This can be considered fractional Fermi coordinates. When applyingan infinite dimensional Lyapunov–Schmidt reduction, the orthogonality conditionis to be expressed in the extension. The essential difference from the classical case[68] is that the inner problem is subdivided into many pieces of size R = o(ε−1),where ε is the scaling parameter, so that the manifold is nearly flat on each piece.In this way, in terms of the Fermi normal coordinates, the equations can be wellapproximated by a model problem.2.1.3 A brief descriptionWe define an approximate solution u∗(x) using the one-dimensional profile in thetubular neighborhood of Mε = {|xn|= Fε(|x′|)}, namely u∗(x)=w(z)where z is thenormal coordinate and Fε is close to the catenoid ε−1 cosh−1(ε|x′|) near the origin.In contrast to the classical case we take into account the non-local interactions nearinfinity and define u∗(x) = w(z+)+w(z−)+1 where z± are the signed distances tothe upper and lower leaves M±ε = {xn =±Fε(|x′|)}. As hinted in Corollary 2.6.3,9Fε(r) ∼ r 22s+1 as r→ +∞. The parts of u∗ are glued to the constant solutions ±1smoothly to the regions where the Fermi coordinates are not well-defined.We look for a real solution of the form u= u∗+ϕ , where ϕ is small and satisfies(−∆)sϕ+ f ′(u∗)ϕ = g. (2.2)Our new idea is to localize the error in the near interface into many pieces of dia-meter R = o(ε−1) for another parameter R which is to be taken large. At eachpiece the hypersurface is well-approximated by some tangent hyperplane. There-fore, using Fermi coordinates, it suffices to study the model problem where u∗(x)is replaced by w(z) in (2.2).As opposed to the local case s = 1, an integration by parts is not available forthe fractional Laplacian in the z-direction, unless n = 1. So we develop a lineartheory using the Caffarelli–Silvestre local extension [43].Finally we will solve a non-local, non-linear reduced equation which takes theform H[Fε ] = O(ε2s−1) for 1 < r ≤ r0,H[Fε ] =Cε2s−1F2sε(1+o(1)) for r > r0,where H[Fε ] denotes the mean curvature of the surface described by Fε . (Notethat the surface is adjusted far away through the nonlocal interactions of the leafs.A similar phenomenon has been observed in Agudelo, del Pino and Wei [10] fors = 1 and dimensions ≥ 4.) A solution of the desired form can be obtained usingthe contraction mapping principle, justifying the a priori assumptions on Fε .In this setting, our main result can be stated as follows.Theorem 2.1.3. Let 1/2 < s < 1 and n = 3. For all sufficiently small ε > 0, thereexists a rotationally symmetric solution u to (2.1) with the zero level set Mε ={(x′,x3) ∈ R3 : |x3|= Fε(|x′|)}, whereFε(r)∼ε−1 cosh−1(εr) for r ≤ rε ,r 22s+1 for r ≥ δ0|logε|rε ,10where rε =( |logε|ε) 2s−12and δ0 > 0 is a small fixed constant.In a forthcoming paper [51], together with Juan Da´vila and Manuel del Pino,we will construct similarly a global minimizer on the Simons’ cone. Via theJerison–Monneau program [120], this provides counter-examples to the De Giorgiconjecture for fractional Allen–Cahn equation in dimensions n≥ 9 for s ∈ (12 ,1).Remark 2.1.4. Our approach depends crucially on the assumption s ∈ (12 ,1).Firstly, it is only in this regime that the local mean curvature alone appears inthe error estimate. A related issue is also seen in the choice of those parametersbetween 0 and (a factor times) 2s− 1. Secondly, it gives the L2 integrabilityof an integral involving the kernel wz in the extension. It will be interestingto see whether this gluing method will work in the case s = 12 under suitablemodifications.On the other hand, we do not know how to deal with other pseudo-differentialoperators which cannot be realized locally.This chapter is organized as follows. We outline the argument with key resultsin Section 2.2. In Section 2.3 we compute the error using an expansion of thefractional Laplacian in the Fermi coordinates. In Section 2.4 we develop a lineartheory and then the gluing reduction is carried out in Section 2.5. Finally in Section2.6 we solve the reduced equation.2.2 Outline of the construction2.2.1 Notations and the approximate solutionLet• s ∈ (12 ,1), α ∈ (0,2s−1), τ ∈(1,1+ α2s),• M be an approximation to the catenoid defined by the function F ,M ={(x′,xn) : |xn|= F(∣∣x′∣∣), ∣∣x′∣∣≥ 1} ,11• ε > 0 be the scaling parameter inMε = ε−1M ={xn = Fε(∣∣x′∣∣) = ε−1F(ε∣∣x′∣∣)} ,• z be the normal coordinate direction in the Fermi coordinates of the rescaledmanifold, i.e. signed distance to the Mε , with z > 0 for xn > F(ε|x′|)> 0,• y+, z+ be respectively the projection onto and signed distance (increasing inxn) from the upper leafM+ε ={xn = Fε(∣∣x′∣∣)} ,• y−, z− be respectively the projection onto and signed distance (decreasing inxn) to the lower leafM−ε ={xn =−Fε(∣∣x′∣∣)} ,• δ¯ > 0 be a small fixed constant so that the Fermi coordinates near Mε isdefined for |z| ≤ 8δ¯ε ,• R¯ > 0 be a large fixed constant,• R0 be the width of the tubular neighborhood of Mε where Fermi coordinatesare used, see (2.3),• R1 be the radius of the cylinder from which the main contribution of (−∆)sis obtained, see Proposition 2.2.1,• R2 > 4R¯ε be the radius of the inner gluing region (i.e. threshold of the end,see Section 2.2.3),• u∗o(x) = sign (xn−Fε(|x′|)) for xn > 0 and is extended continuously (i.e.u∗o(x) = +1 for |x′| ≤ ε−1),• η : R→ [0,1] be a cut-off with η = 1 on (−∞,1] and η = 0 on [2,+∞),• χ : R→ [0,1] be a cut-off with χ = 0 on (−∞,0] and χ = 1 on [1,+∞),• ‖κ‖α (0≤ α < 1) be the Ho¨lder norm of the curvature, see Lemma 2.3.6,12• 〈x〉=√1+ |x|2.Define the approximate solutionu∗(x) = η(ε|z|δ¯R0(|x′|))(w(z)+χ(∣∣x′∣∣− R¯ε)(w(z+)+w(z−)+1−w(z)))+(1−η(ε|z|δ¯R0(|x′|)))u∗o(x),(2.3)whereR0 = R0(∣∣x′∣∣) = 1+χ (∣∣x′∣∣− R¯)(F2sε (∣∣x′∣∣)−1) .Roughly,• u∗(x) = +1 for large |z|, small |x′| and large |xn|,• u∗(x) =−1 for large |z|, large |x′| and small |xn|,• u∗(x) = w(z) for small |z| and small |x′|,• u∗(x) = w(z+)+w(z−)+1 for small |z| and large |x′|.The main contributions of (−∆)s come from the inner region with certain ra-dius. We choose such radius that joins a small constant times ε−1 to a power of Fεas |x′| increases. More precisely, let us setR1 = R1(∣∣x′∣∣) = η(∣∣x′∣∣− 2R¯ε+2)δ¯ε+(1−η(∣∣x′∣∣− 2R¯ε+2))Fτε (∣∣x′∣∣),(2.4)where τ ∈ (1,1+ α2s). We remark that the factor 2 is inserted to make sure thatu∗(x) = w(z+) +w(z−)− 1 in the whole ball of radius Fτε (|x′|) where the mainorder terms of (−∆)su∗ are obtained.2.2.2 The errorDenote the error by S(u∗)= (−∆)su∗+(u∗)3−u∗. In a tubular neighborhood wherethe Fermi coordinates are well-defined, write x = y+ zν(y) where y = y(|x′|) =13(|x′|,Fε(|x′|)) ∈Mε and ν(y) = ν(y(|x′|)) = (−DFε(|x′|),1)√1+ |DFε(|x′|)|2be the unit normalpointing up in the upper half space (and down in the lower half).Proposition 2.2.1. Let x = y+ zν(y) ∈ Rn. If |z| ≤ R1, where R1 as in (2.4), thenwe haveS(u∗)(x) =cH(z)HMε (y)+O(ε2s), for1ε≤ r ≤ 4R¯ε,cH(z+)HM+ε (y+)+ cH(z−)HM−ε (y−)+3(w(z+)+w(z−))(1+w(z+))(1+w(z−))+O(F−2sτε), for r ≥ 4R¯ε.The proof is given in Section 2.3.2.2.3 The gluing reductionWe look for a solution of (2.1) of the form u = u∗+ϕ so that(−∆)sϕ+ f ′(u∗)ϕ = S(u∗)+N(ϕ) in Rn,where N(ϕ) = f (u∗+ϕ)− f (u∗)− f ′(u∗)ϕ . Consider the partition of unity1 = η˜o+ η˜++ η˜−+i¯∑i=1η˜i,where the support of each η˜i is a region of radius R centered at some yi ∈Mε , andη˜± are supported on a tubular neighborhood of the ends of M±ε respectively. It willbe convenient to denote I = {1, . . . , i¯} andJ =I ∪{+,−}. For j ∈J , let ζ jbe cut-off functions such that the sets{ζ j = 1}include supp η˜ j, with comparablespacing that is to be made precise. We decomposeϕ = φo+ζ+φ++ζ−φ−+i¯∑i=1ζiφi = φo+ ∑j∈Jζ jφ j,in which14• φo solves for the contribution of the error away from the interface (supportof η˜o),• φ± solves for that in the far interfaces near M±ε (support of η˜±),• φi solves for that in a compact region near the manifold (support of η˜i).In the following we write ∆(y,z) = ∆y+∂zz.We consider the approximate linear op-erators Lo = (−∆)s+2 for φo,L = (−∆(y,z))s+ f ′(w) for φ j, j ∈J .Notice that w is not exactly the approximate solution in the far interface. We re-arrange the equation as(−∆)s(φo+ ∑j∈Jζ jφ j)+ f ′(u∗)(φo+ ∑j∈Jζ jφ j)= S(u∗)+N(ϕ),Loφo+ζ+Lφ++ζ−Lφ−+i¯∑i=1ζiLφi=(η˜o+ η˜++ η˜−+i¯∑i=1η˜i)·(S(u∗)+N(ϕ)+(2− f ′(u∗))φo− ∑j∈J[(−∆(y,z))s,ζ j]φ j+ ∑j∈Jζ j( f ′(w j)− f ′(u∗))φ j− ∑j∈J((−∆x)s− (−∆(y,z))s)(ζ jφ j)), (2.5)where [(−∆(y,z))s,ζ j]φ j = (−∆(y,z))s(ζ jφ j)− ζ j(−∆(y,z))sφ j, and the summands inthe last term means(−∆x)s(ζ jφ j)(Yj(y)+ zν(Yj(y)))− (−∆(y,z))s(η¯ jζ¯ φ¯(y,z))for ζ j = η¯ j(y)ζ¯ (z) and φ j(Yj(y) + zν(Yj(y))) = φ¯ j(y,z) with a chart y = Yj(y)of Mε . In fact, for j ∈ I one can parameterize Mε locally by a graph15over a tangent hyperplane, and for j ∈ {+,−} one uses the natural graphM±ε = {(y,±Fε(|y|)) : |y| ≥ R2}.Let us denote the last bracket of the right hand side of (2.5) by G . Since η˜ j =ζ jη˜ j, we will have solved (2.5) if we get a solution to the systemLoφo = η˜oG for x ∈ Rn,Lφ¯+ = η˜+G for (y,z) ∈ Rn−1×R,Lφ¯− = η˜−G for (y,z) ∈ Rn−1×R,Lφ¯i = η˜iG for (y,z) ∈ Rn−1×R,for all i ∈I . Except the outer problem with Lo = (−∆)s+2, the linear operator Lin all the other equations has a kernel w′ and so we will use an infinite dimensionalLyapunov–Schmidt reduction procedure.From now on we consider the product cut-off functions, defined in the Fermicoordinates (y,z) where y= Y (y) is given by a chart of Mε ,η˜ j(x) = η j(y)ζ (z), for j ∈J .The diameters of ζ (z) and ηi(y) are of order R, a parameter which we chooseto be large (before fixing ε). We may assume, without loss of generality, thatfor i ∈ I , ηi(y) is centered at yi ∈ Mε , BR(yi) ⊂ {η˜i = 1} ⊂ supp η˜i ⊂ B2R(yi),|Dη˜i|= O(R−1), and |yi1−yi2 |R ≥ c > 0 for any i1, i2 ∈I .We define the projection orthogonal to the kernels w′(z),Πg(y,z) = g(y,z)− c(y)w′(z), c(y) =∫Rζ (z˜)g(y, z˜)w′(z˜)dz˜∫Rζ (z˜)w′(z˜)2 dz˜.Note that in the region of integration |z| ≤ 2R < δ¯ ε−1 the Fermi coordinates arewell-defined, and that the projection is independent of j ∈J .We define the norm‖φ‖µ,σ = sup(y,z)∈Rn〈y〉µ 〈z〉σ |φ(y,z)|,16where 〈y〉=√1+ |y|2. Motivated by Proposition 2.2.1 and Lemma 2.4.6, for eachi ∈I we expect the decay∥∥φ¯i(y,z)∥∥µ,σ ≤CRµ+σ 〈yi〉− 4s2s+1 .So we define‖φi‖i,µ,σ = 〈yi〉θ∥∥φ¯i∥∥µ,σ = 〈yi〉θ sup(y,z)∈Rn〈y〉µ 〈z〉σ ∣∣φ¯i(y,z)∣∣,with 1 < θ < 1+ 2s−12s+1 =4s2s+1 < 2s. At the ends M±ε where r ≥ R2 we have, forµ < 4s2s+1 −θ , ∥∥φ¯±(y,z)∥∥µ,σ ≤CR−( 4s2s+1−µ)2 .This suggests‖φ±‖±,µ,σ = Rθ2∥∥φ¯±∥∥µ,σ = Rθ2 sup(y,z)∈Rn〈y〉µ 〈z〉σ ∣∣φ¯±(y,z)∣∣,with 0 < θ < 2s−12s+1 −µ . Therefore for j ∈J , we consider the Banach spacesX j ={φ j :∥∥φ j∥∥ j,µ,σ < C˜δ} ,where, under the constraint R≤ |logε|, δ = δ (R,ε) = Rµ+σε 4s2s+1−θ with 1 < θ <1+ 2s−12s+1 =4s2s+1 . For the other parameters we take 0 < µ <4s2s+1 − θ < θ suffi-ciently small and R2 sufficiently large, so that Rµ2 δ is small and 2−2s<σ < 2s−µ .The decay of order σ > 2−2s in the z-direction will be required in the orthogonal-ity condition (2.21). That Rµ2 δ is small will be used in the inner gluing reduction.The condition σ +µ < 2s ensures that the contribution of the term (2− f ′(u∗))φois small compared to S(u∗).We will first solve the outer equation for φo. Let us writeMε,R = {y+ zν(y) : y ∈Mε and |z|< R}for the tubular neighborhood of Mε with width R.17Proposition 2.2.2. Consider‖φo‖θ = sup(x′,xn)∈Rn〈x′〉θ 〈dist(x,Mε,R)〉2s |φo(x)|,Xo ={φo : ‖φo‖θ ≤ C˜εθ}.If φ j ∈ X j for all j ∈J with sup j∈J∥∥φ j∥∥ j,µ,σ ≤ 1, then there exists a uniquesolution φo =Φo((φ j) j∈J ) toLoφo = η˜oG = η˜o(S(u∗)+N(ϕ)+(2− f ′(u∗))φo− ∑j∈J[(−∆(y,z))s,ζ j]φ j+ ∑j∈Jζ j( f ′(w j)− f ′(u∗))φ j− ∑j∈J((−∆x)s− (−∆(y,z))s)(ζ jφ j))in Rn (2.6)in Xo such that for any pairs (φ j) j∈J and (ψ j) j∈J in the respective X j withsup j∈J∥∥φ j∥∥ j,µ,σ ≤ 1,∥∥Φo((φ j) j∈J )−Φo((ψ j) j∈J )∥∥θ ≤Cεθ supj∈J∥∥φ j−ψ j∥∥ j,µ,σ . (2.7)The proof is carried out in Section 2.5.2.Then the equationsLφ¯ j(y,z) = η j(y)ζ (z)G (y,z)are solved in two steps: (1) eliminating the part of error orthogonal to the kernels,i.e.Lφ¯ j(y,z) = η j(y)ζ (z)ΠG (y,z); (2.8)and (2) adjust Fε(r) such that c(y) = 0, i.e. to solve the reduced equation∫Rζ (z)G (y,z)w′(z)dz = 0. (2.9)Using the linear theory in Section 2.4, step (1) is proved in the following18Proposition 2.2.3. Suppose µ ≤ θ . Then there exists a unique solution (φ j) j∈J ,φ j ∈ X j, to the systemLφ¯ j = η˜ jΠG = η jζΠ(S(u∗)+N(ϕ)+(2− f ′(u∗))φo− ∑j∈J[(−∆(y,z))s,ζ j]φ j+ ∑j∈Jζ j( f ′(w j)− f ′(u∗))φ j− ∑j∈J((−∆x)s− (−∆(y,z))s)(ζ jφ j))(2.10)for (y,z) ∈ Rn.The proof is given in Section 2.5.3.Step (2) is outlined in the next subsection.2.2.4 Projection of error and the reduced equationAs shown above, the error is to be projected onto w′j weighted with a cut-off func-tion ζ supported on [−2R,2R]. In fact we haveProposition 2.2.4 (The reduced equation). In terms of the rescaled function F(r)=εFε(ε−1r) and its inverse r=G(z)where G : [0,+∞)→ [1,+∞), (2.9) is equivalentto the systemHM(G(z),z) = G′(z)√1+G′(z)2′− 1G(z)√1+G′(z)2= N1[F ] for 0≤ z≤ z1,HM(r,F(r)) =1r rF ′(r)√1+F ′(r)2′ = N1[F ] for r1 ≤ r ≤ 4R¯,F ′′(r)+F ′(r)r− C¯0ε2s−1F2s(r)= N2[F ] for r ≥ 4R¯,(2.11)19subject to the boundary conditionsG(0) = 1G′(0) = 0F(r1) = z1F ′(r1) =1G′(z1),(2.12)where z1 = F(r1) = O(1), N1[F ] = O(ε2s−1) and N2[F ] = o(ε2s−1F2s0 (r)), with F0 asin Corollary 2.6.3. Moreover, N1 and N2 have a Lipschitz dependence on F.This is proved in Section 2.6.1.The equation (2.11)–(2.12) is to be solved in a space with weighted Ho¨ldernorms allowing sub-linear growth. More precisely, for any α ∈ (0,1), γ ∈ R wedefine the norms‖φ‖∗ = sup[r1,+∞)rγ−2|φ(r)|+ sup[r1,+∞)rγ−1∣∣φ ′(r)∣∣+ sup[r1,+∞)rγ∣∣φ ′′(r)∣∣+ supr 6=ρ in [r1,+∞)min{r,ρ}γ+α |φ′′(r)−φ ′′(ρ)||r−ρ|α (2.13)and‖h‖∗∗ = supr∈[1,+∞)rγ |h(r)|+ supr 6=ρ in [1,+∞)min{r,ρ}γ+α |h(r)−h(ρ)||r−ρ|α . (2.14)Proposition 2.2.5. There exists a solution to (2.11) in the spaceX∗ =(G,F) ∈C2,α([0,z1])×C2,αloc ([r1,+∞)) :‖G‖C2,α ([0,z1]) <+∞,‖F‖∗ <+∞,(2.12) holds .The proof is contained in Section 2.6.202.3 Computation of the error: Fermi coordinatesexpansionWe prove the followingProposition 2.3.1 (Expansion in Fermi coordinates). Suppose 0< α < 2s−1 andFε ∈C2,αloc ([1,+∞)). Let x0 = y0+z0ν(y0) where y0 = (x′,Fε(|x′|)) is the projectionof x0 onto Mε , and u0(x) = w(z). Then for any τ ∈(1,1+ α2s)and |z0| ≤ R1, wehave(−∆)su0(x0) = w(z0)−w(z0)3+ cH(z0)HMε (y0)+N1[F ]wherecH(z0) =C1,s∫Rw(z0)−w(z)|z0− z|1+2s(z0− z)dz,R1 = R1(∣∣x′∣∣) = η(∣∣x′∣∣− 2R¯ε+2)δ¯ε+(1−η(∣∣x′∣∣− 2R¯ε+2))Fτε (∣∣x′∣∣),and N1[F ] = O(R−2s1)is finite in the norm ‖·‖∗∗.Remark 2.3.2. cH(z0) is even in z0. AlsocH(z0) =C1,s2s−1∫Rw′(z)|z0− z|2s−1dz∼ 〈z0〉−(2s−1) .This implies Proposition 2.2.1. A proof is given at the end of this section.A similar computation gives the decay in r = |x′| away from the interface.Corollary 2.3.3. Suppose x0 = y0+ z0ν(y0), y0 = (x′0,Fε(r0)) and z0 ≥ cr22s+10 .(−∆)su∗(x0) = O(r− 4s2s+10)as r0→+∞.Proof. Take a ball around x0 of radius of order r22s+10 . In the inner region one usesthe closeness to +1 of the approximate solution u∗.For more general functions one has a less precise expansion. On compact sets,we have21Corollary 2.3.4. Let u1(x) = φ(y,z) in a neighborhood of x0 = y0+z0ν(y0) where|y0|, |z0| ≤ 4R = o(ε−1), and u1 = 0 outside a ball of radius 8R. Then(−∆x)su1(x0) = (−∆(y,z))sφ(y0,z0) · (1+O(R‖κ‖0))+O(R−2s1(|φ(y0,z0)|+ sup|(y0−y,z0−z)|≥R1|φ(y,z)|)).Proof. The lower order terms contain either κi|z0| or κi|y0|, where i = 1 or 2.At the ends of the catenoidal surface we need the followingCorollary 2.3.5. Let u1(x) = φ(y,z) in a neighborhood of x0 = y0+ z0ν(y0) where|y0| ≥ R2, |z0| ≤ 4R = o(ε−1), and u1 = 0 when z≥ 8R. Then(−∆x)su1(x0) = (−∆(y,z))sφ(y0,z0) ·(1+O(F−(2s−τ)ε))+O(F−2sτε(|φ(y0,z0)|+ sup|(y0−y,z0−z)|≥Fτε|φ(y,z)|)).To prove Proposition 2.3.1, we consider Mε as a graph in a neighborhood of y0over its tangent hyperplane and use the Fermi coordinates. Suppose (y1,y2,z) is anorthonormal basis of the tangent plane of Mε at y0. WriteCR1 ={(y,z) ∈ R2×R : |y| ≤ R1, |z| ≤ R1}.Then there exists a smooth function g : BR1(0)→R such that, in the (y,z) coordin-ates,Mε ∩CR1 ={(y,g(y)) ∈ R3 : |y| ≤ R1}.Then g(0) = 0, Dg(0) = 0 and ∆g(0) = 2HMε (x0). We may also assumethat ∂y1y2g(0) = 0. We denote the principal curvatures at y by κi(y) so thatκi(0) = ∂yiyig(0).We state a few lemmata whose non-trivial proofs are postponed to the end ofthis section.22Lemma 2.3.6 (Local expansions). Let |y| ≤ R1. For i = 1,2 we have|κi(y)−κi(0)|. ‖κi‖Cα (B2R1 (|x′|)) |y|α .∥∥F−2sε ∥∥Cα (B1(|x′|)) |y|α.ε2s+α |y|α for all |x′| ≤ 2R¯ε,F−2sε (|x′|)|x′|α |y|α for all |x′| ≥ R¯ε.The quantity ‖Fε‖C2,α (BR1 (|x′|) .∥∥F−2sε ∥∥Cα (B1(|x′|)) will be used repeatedly and willbe simply denoted by ‖κ‖α , as a function of |x′|, for any 0≤ α < 1. We haveg(y) =122∑i=1κi(0)y2i +O(‖κ‖α |y|2+α),Dg(y) · y =2∑i=1κi(0)y2i +O(‖κ‖α |y|2+α),|Dg(y)|2 = O(‖κ‖20 |y|2).In particular,g(y)−Dg(y) · y =−122∑i=1κi(0)y2i +O(‖κ‖α |y|2+α)= O(‖κ‖0 |y|2),√1+ |Dg(y)|2−1 = O(‖κ‖20 |y|2),1− 1√1+ |Dg(y)|2= O(‖κ‖20 |y|2),g(y)2 = O(‖κ‖20 |y|4).Lemma 2.3.7 (The change of variable). Let |y|, |z|, |z0| ≤ R1. Under the Fermichange of variable x =Φ(y,z) = y+ zν(y), the Jacobian determinantJ(y,z) =√1+ |Dg(y)|2(1+κ1(y)z)(1+κ2(y)z)23satisfiesJ(y,z) = 1+(κ1(0)+κ2(0))z+O(‖κ‖α |y|α |z|)+O(‖κ‖20 (|y|2+ |z|2)) ,and the kernel |x0− x|−3−2s has an expansion|x0− x|−3−2s = |(y,z0− z)|−3−2s[1+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|2+α(|z|+ |z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|2(|y|2+ |z|2+ |z0|2)|(y,z0− z)|2)].Lemma 2.3.8 (Reducing the kernel). There holdC3,s∫R21|(y,z0− z)|3+2sdy =C1,s1|z0− z|1+2s,C3,s∫R2y2i|(y,z0− z)|5+2sdy =13+2sC1,s1|z0− z|1+2sfor i = 1,2,∫R2|y|α|(y,z0− z)|3+2sdy =C1|z0− z|1+2s−α.Proof of Proposition 2.3.1. The main contribution of the fractional Lapla-cian comes from the local term which we compute in Fermi coordinatesΦ(y,z) = y+ zν(y),(−∆)su0(x0) =C3,s∫Φ(CR1 )u0(x0)−u0(x)|x− x0|3+2sdx+O(R−2s1 )=C3,s∫∫CR1w(z0)−w(z)|Φ(y0,z0)−Φ(y,z)|3+2sJ(y,z)dydz+O(R−2s1 ).By Lemma 2.3.7 we haveJ(y,z) = 1+(κ1(0)+κ2(0))z+O(‖κ‖α |y|α |z|)+O(‖κ‖20 (|y|2+ |z|2)) ,241|Φ(y0,z0)−Φ(y,z)|3+2s=1|(y,z0− z)|3+2s[1+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|2+α(|z|+ |z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|2(|y|2+ |z|2+ |z0|2)|(y,z0− z)|2)].HenceJ(y,z)|Φ(y0,z0)−Φ(y,z)|3+2s=1|(y,z0− z)|3+2s·[1+(κ1(0)+κ2(0))z+O(‖κ‖α |y|α |z|)+O(‖κ‖20 (|y|2+ |z|2))]·[1+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|2+α(|z|+ |z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|2(|y|2+ |z|2+ |z0|2)|(y,z0− z)|2)]=1|(y,z0− z)|3+2s[1+(κ1(0)+κ2(0))z+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|α(|z|+ |z0|))+O(‖κ‖20 (|y|2+ |z|2+ |z0|2))].25We have(−∆)su0(x0)=C3,s∫∫CR1w(z0)−w(z)|Φ(y0,z0)−Φ(y,z)|3+2sJ(y,z)dydz+O(R−2s1 )=C3,s∫∫CR1w(z0)−w(z)|(y,z0− z)|3+2s[1+(κ1(0)+κ2(0))z+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|α(|z|+ |z0|))+O(‖κ‖20 (|y|2+ |z|2+ |z0|2))]= I1+ I2+ I3+ I4+ I5.whereI1 =C3,s∫∫CR1w(z0)−w(z)|(y,z0− z)|3+2sdydz,I2 =C3,s(κ1(0)+κ2(0))∫∫CR1w(z0)−w(z)|(y,z0− z)|3+2szdydz,I3 =C3,s3+2s22∑i=1κi(0)∫∫CR1w(z0)−w(z)|(y,z0− z)|5+2s(z0+ z)y2i dydz,I4 = O(‖κ‖α)∫∫CR1∣∣∣w(z0)−w(z)−χB11(z0)(z)w′(z0)(z0− z)∣∣∣|(y,z0− z)|3+2s|y|α· (|z|+ |z0|)dydz,I5 = O(‖κ‖20)∫∫CR1∣∣∣w(z0)−w(z)−χB11(z0)(z)w′(z0)(z0− z)∣∣∣|(y,z0− z)|3+2s· (|y|2+ |z|2+ |z0|2)dydz.In the last terms I4 and I5, the linear odd term near the origin has been added toeliminate the principal value before the integrals are estimated by their absolutevalues. One may obtain more explicit expressions by extending the domain andusing Lemma 2.3.8 as follows. I1 resembles the fractional Laplacian of the one-26dimensional solution.I1 =C3,s∫∫R3w(z0)−w(z)|(y,z0− z)|3+2sdydz−C3,s∫∫R3\CR1w(z0)−w(z)|(y,z0− z)|3+2sdydz=C3,s∫R(w(z0)−w(z))∫R21|(y,z0− z)|3+2sdydz+O(∫ ∞R1ρ−3−2sρ2 dρ)=C1,s∫Rw(z0)−w(z)|z0− z|1+2sdz+O(R−2s1)= w(z0)−w(z0)3+O(R−2s1).Hereafter ρ =√|y|2+ |z0− z|2. I2 and I3 are of the next order where we see themean curvature.I2 =−C3,s2∑i=1κi(0)∫∫CR1w(z0)−w(z)|(y,z0− z)|3+2szdydz=−C3,s2∑i=1κi(0)∫∫R3w(z0)−w(z)|(y,z0− z)|3+2szdydz−C3,s2∑i=1κi(0)∫∫R3\CR1w(z0)−w(z)|(y,z0− z)|3+2s(z0+(z− z0))dydz=−C1,s2∑i=1κi(0)∫Rw(z0)−w(z)|z0− z|1+2szdz+O(‖κ‖0 |z0|∫ ∞R11ρ3+2sρ2 dρ)+O(‖κ‖0∫ ∞R1ρρ3+2sρ2 dρ)=−2(C1,s∫Rw(z0)−w(z)|z0− z|1+2szdz)HMε (y0)+O(‖κ‖0 R−2s1 (|z0|+R1)) .27Also,I3 =C3,s3+2s22∑i=1κi(0)∫∫R3w(z0)−w(z)|(y,z0− z)|5+2s(z0+ z)y2i dydz+O(‖κ‖0)∫∫R3\CR1w(z0)−w(z)|(y,z0− z)|5+2s(2z0− (z0− z))y2i dydz=C1,s122∑i=1κi(0)∫Rw(z0)−w(z)|z0− z|1+2s(z0+ z)dz+O(‖κ‖0 |z0|∫ ∞R1ρ2ρ5+2sρ2 dρ)+O(‖κ‖0∫ ∞R1ρ3ρ5+2sρ2 dρ)=(C1,s∫Rw(z0)−w(z)|z0− z|1+2s(z0+ z)dz)HMε (y0)+O(‖κ‖0 R−2s1 (|z0|+R1)) .The remainder terms I4 and I5 are estimated as follows.I4 = O(‖κ‖α)∫∫CR1∣∣∣w(z0)−w(z)−χB11(z0)(z)w′(z0)(z0− z)∣∣∣|(y,z0− z)|3+2s|y|α(|z|+ |z0|)dydz= O(‖κ‖α)∫R∣∣∣w(z0)−w(z)+χB11(0)(z)w′(z0)(z0− z)∣∣∣·∫R2|y|α(|z0− z|+ |z0|)(|y|2+ |z0− z|2) 3+2s2dydz+O(‖κ‖α (|z|+ |z0|)∫ ∞R1ραρ3+2sρ2 dρ)= O(‖κ‖α)∫R∣∣∣w(z0)−w(z)+χB11(0)(z)w′(z0)(z0− z)∣∣∣|z0− z|1+2s−α(|z0− z|+ |z0|) dz+O(‖κ‖α R−2s+α1 (|z|+ |z0|))= O(‖κ‖α (1+R−2s+α1 (|z|+ |z0|))) .28I5 = O(‖κ‖20)∫∫CR1∣∣∣w(z0)−w(z)−χB11(z0)(z)w′(z0)(z0− z)∣∣∣|(y,z0− z)|3+2s· (|y|2+ |z|2+ |z0|2)dydz= O(‖κ‖20)(1+∫ R11ρ2+ |z0|2ρ3+2sρ2 dρ)= O(‖κ‖20 (1+R2−2s1 +R−2s1 |z0|2)).In conclusion, we have, since |z0| ≤ R1 and α < 2s−1,(−∆)su0(x0) = w(z0)−w(z0)3+(C1,s∫Rw(z0)−w(z)|z0− z|1+2s(z0− z)dz)HMε (y0)+O(R−2s1(1+‖κ‖0 R1+‖κ‖α R2s1 +‖κ‖20 R21))= w(z0)−w(z0)3+ cH(z0)HMε (y0)+O(R−2s1 ),the last line following from the estimate‖κ‖α R2s1 .εα for |x′| ≤ 2R¯εF2s(τ−1)ε|x′|α for |x′| ≥ R¯ε.εα for |x′| ≤ 2R¯εεα−2s(τ−1)(ε|x′|)−2s(τ−1)(1− 22s+1 ) for |x′| ≥ R¯ε. εα−2s(τ−1).The finiteness of the remainder in the norm ‖·‖∗∗ is a tedious but straightforwardcomputation. As an example, the difference of the exterior error with two radii Fτεand Gτε is controlled by∣∣∣∣∣∫Φ(CcFτε)u0(x0)−u0(x)|x− x0|3+2sdx−∫Φ(CcGτε)u0(x0)−u0(x)|x− x0|3+2sdx∣∣∣∣∣=∣∣∣∣∣∫∫CGτε \CFτεw(z0)−w(z)|Φ(y0,z0)−Φ(y,z)|3+2sJ(y,z)dydz∣∣∣∣∣.29Following the computations in the above proof, a typical term would beO(G−2sτε −F−2sτε)= O(r−2(2sτ+1)2s+1 |Fε −Gε |),which implies Lipschitz continuity with decay in r.Similarly we prove the expansion at the end.Proof of Corollary 2.3.5. We recall that a tubular neighborhood of an end of M+εare parameterized byx = y+ zν(y) = (y,Fε(r))+ z(−F ′ε(r) yr ,1)√1+F ′ε(r)2for r = |y|> r0, |z|< δ¯ε ,where r = |y|. In place of Lemma 2.3.7 we have for |z| ≤ Fτε (r) with 1< τ < 2s+12 ,J(y,z) =(1+O(F ′ε(r)2))(1+O(F ′′ε (r)Fτε (r)))2=(1+O(F−(2s−1)ε (r)))(1+O(F−(2s−τ)ε (r)))2= 1+O(F−(2s−τ)ε (r)),|x− x0|2 =(|y0− y|2+ |z0− z|2)(1+O(Fτε (r)F′′ε (r)))=(|y0− y|2+ |z0− z|2)(1+O(F−(2s−τ)ε)).The result follows by the same proof as in Proposition 2.3.1.We now give a proof of the error estimate stated in Section 2.2.Proof of Proposition 2.2.1. Using the Fermi coordinates expansion of the frac-tional Laplacian (Proposition 2.3.1), we have, in an expanding neighborhood ofMε , the following estimates on the error:• For 1ε≤ |x′| ≤ 2R¯εand |z| ≤ δ¯ε,S(u∗)(x) = cH(z)HMε (y)+O(ε2s).30• For |x′| ≥ 4R¯εand |z| ≤ Fτε (|x′|),S(u∗)(x) = (−∆)s(w(z+)+w(z−)+1)+ f (w(z+)+w(z−)−1)+O(F−2sτε)= f (w(z+)+w(z−)+1)− f (w(z+))− f (w(z−))+ cH(z+)HM+ε (y+)+ cH(z−)HM−ε (y−)+O(F−2sτε)= 3(w(z+)+w(z−))(1+w(z+))(1+w(z−))+ cH(z+)HM+ε (y+)+ cH(z−)HM−ε (y−)+O(F−2sτε).• For 2R¯ε≤ |x′| ≤ 4R¯ε, xn > 0 and |z| ≤ R1(|x′|),S(u∗)(x) = (−∆)sw(z+)+(−∆)s((1−η(∣∣x′∣∣− R¯ε)(w(z−)+1)))+ f(w(z+)+(1−η(∣∣x′∣∣− R¯ε)(w(z−)+1)))= cH(z+)HMε (y+)+O(ε2s).Here the second term is small because of the smallness of the cut-off errorup to two derivatives.• For 2R¯ε≤ |x′| ≤ 4R¯ε, xn < 0 and |z| ≤ R1(|x′|), we have similarlyS(u∗)(x) = cH(z−)HMε (y−)+O(ε2s).This completes the proof.31Proof of Lemma 2.3.7. Referring to Lemma 2.3.6 and keeping in mind that‖κ‖0 R1 = o(1), for the Jacobian determinant we haveJ(y,z) = 1+(κ1(0)+κ2(0))z+((κ1+κ2)(y)− (κ1+κ2)(0))z+(√1+ |Dg(y)|2−1)(1+(κ1(y)+κ2(y))z+κ1(y)κ2(y)z2)= 1+(κ1(0)+κ2(0))z+O(‖κ‖α |y|α |z|)+O(‖κ‖20 |z|2)+O(‖κ‖20 |y|2)(1+O(‖κ‖0 |z|))2= 1+(κ1(0)+κ2(0))z+O(‖κ‖α |y|α |z|)+O(‖κ‖20 (|y|2+ |z|2)) .To expand the kernel we first considerx0− x = (y,g(y))− (0,z0)+ z (−Dg(y),1)√1+ |Dg(y)|2,|x0− x|2= |y|2+g(y)2+ z2+ z20−2zz0√1+ |Dg(y)|2+2z(g(y)−Dg(y) · y)√1+Dg(y)2−2z0g(y)= |y|2+ |z0− z|2+2z(g(y)−Dg(y) · y)−2z0g(y)+g(y)2+(2zz0−2z(g(y)−Dg(y) · y))1− 1√1+ |Dg(y)|2= |(y,z0− z)|2− (z0+ z)2∑i=1κi(0)y2i +O(‖κ‖α |y|2+α(|z|+ |z0|))+O(‖κ‖20 |y|4)+O(‖κ‖20 |y|2|z|(|z0|+‖κ‖0 |y|2))= |(y,z0− z)|2− (z0+ z)2∑i=1κi(0)y2i+O(‖κ‖α |y|2+α(|z|+ |z0|))+O(‖κ‖20 |y|2(|y|2+ |z||z0|)).32By binomial theorem,|x0− x|−3−2s= |(y,z0− z)|−3−2s[1+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|2+α(|z|+ |z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|2(|y|2+ |z||z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|4(|z0|2+ |z|2)|(y,z0− z)|4)]= |(y,z0− z)|−3−2s[1+3+2s2(z0+ z)2∑i=1κi(0)y2i|(y,z0− z)|2+O(‖κ‖α |y|2+α(|z|+ |z0|)|(y,z0− z)|2)+O(‖κ‖20 |y|2(|y|2+ |z|2+ |z0|2)|(y,z0− z)|2)].Proof of Lemma 2.3.8. The first and third equalities follow by the change of vari-able y = |z0− z|y˜. To prove the second one, we have∫R2y2i|(y,z0− z)|5+2sdy=12∫R2(|y|2+ |z0− z|2)−|z0− z|2(|y|2+ |z0− z|2) 5+2s2dy=12∫R2dy(|y|2+ |z0− z|2) 3+2s2− 12|z0− z|2∫R2dy(|y|2+ |z0− z|2) 5+2s2=12C1,sC3,s1|z0− z|1+2s− 12C3,sC5,s|z0− z|2|z0− z|3+2s=12C1,sC3,s(1− C23,sC1,sC5,s)1|z0− z|1+2s.33Recalling thatCn,s =22ssΓ(1− s)Γ(n+2s2)pi n2,we have1− C23,sC1,sC5,s= 1− Γ(3+2s2)2Γ(1+2s2)Γ(5+2s2) = 1− 1+2s3+2s=23+2sand hence ∫R2y2i|(y,z0− z)|5+2sdy =13+2sC1,sC3,s1|z0− z|1+2s.2.4 Linear theoryIn this section we use a different notation. We write w = w(z, t) for the layer in theextension and w(z) for its trace.2.4.1 Non-degeneracy of one-dimensional solutionConsider the linearized equation of (−∆)su+ f (u) = 0 at w, the one-dimensionalsolution, namely(−∆)sφ + f ′(w)φ = 0 for (y,z) ∈ Rn, (2.15)or the equivalent extension problem (here a = 1−2s)∇ · (ta∇φ) = 0 for (y,z, t) ∈ Rn+1+ta∂φ∂ν+ f ′(w)φ = 0 for (y,z) ∈ Rn.(2.16)Given ξ ∈ Rn−1, we define onX = H1(R2+, ta)34the bilinear form(u,v)X =∫R2+ta(∇u ·∇v+ |ξ |2uv)dzdt+∫Rf ′(w)uvdz.Lemma 2.4.1 (An inner product). Suppose ξ 6= 0. Then (·, ·)X defines an innerproduct on X.Proof. Clearly (u,u)X <∞ for any u ∈ X . For R> 0, denote B+R = BR(0)∩R2+ andits boundary in R2+ by ∂B+R . It suffices to prove that∫B+Rta|∇u|2 dzdt+∫∂B+Rf ′(w)u2 dz =∫B+Rtaw2z∣∣∣∣∇( uwz)∣∣∣∣2 dzdt. (2.17)Since the right hand side is non-negative, the result follows as we take R→ +∞.To check the above equality, we compute∫B+Rtaw2z∣∣∣∣∇( uwz)∣∣∣∣2 dzdt=∫B+Rta∣∣∣∣∇u− uwz∇wz∣∣∣∣2 dzdt=∫B+Rta|∇u|2 dzdt+∫B+Rtau2w2z|∇wz|2 dzdt−∫B+Rta∇(u2) · ∇wzwzdzdt.Since ∇ · (ta∇wz) = 0 in R2+, we can integrate the last integral by parts as−∫B+Rta∇(u2) · ∇wzwzdzdt =−∫∂B+Ru2ta∂νwzwzdz+∫B+Ru2∇ ·(ta∇wzwz)dzdt=∫∂B+Ru2f ′(w)wzwzdz+∫B+Rtau2∇wz ·∇ · 1wz dzdt=∫∂B+Rf ′(w)u2 dz−∫B+Rtau2w2z|∇wz|2 dzdt.Therefore, (2.17) holds and the proof is complete.35Lemma 2.4.2 (Solvability of the linear equation). Suppose ξ 6= 0. For any g ∈C∞c (R2+) and h ∈C∞c (R), there exists a unique u ∈ X of−∇ · (ta∇u)+ ta|ξ |2u = g in R2+ta∂u∂ν+ f ′(w)u = h on ∂R2+.(2.18)Proof. This equation has the weak formulation(u,v)X =∫R2+ta(∇u ·∇v+ |ξ |2uv)dzdt+∫Rf ′(w)uvdz=∫R2+gvdzdt+∫Rhvdz.By Riesz representation theorem, there is a unique solution u ∈ X .Lemma 2.4.3 (Non-degeneracy in one dimension [82, Lemma 4.2]). Let w(z) bethe unique increasing solution of(−∂zz)sw+ f (w) = 0 in R.If φ(z) is a bounded solution of(−∂zz)sφ + f ′(w)φ = 0 in R,then φ(z) =Cw′(z).Lemma 2.4.4 (Non-degeneracy in higher dimensions). Let φ(y,z, t) be a boundedsolution of∇(y,z,t) · (ta∇(y,z,t)φ) = ta(∂tt +at∂t +∂zz+∆y)φ = 0 in Rn+1+ta∂φ∂ν+ f ′(w)φ = 0 on ∂Rn+1+ ,(2.19)36where w(z, t) is the one-dimensional solution so that∇(z,t) · (ta∇(z,t)wz) = ta(∂tt +at∂t +∂zz)wz = 0 in R2+ta∂wz∂ν+ f ′(w)wz = 0 on ∂R2+.Then φ(y,z, t) = cwz(z, t) for some constant c.Proof. For each (z, t)∈R2+, letψ(ξ ,z, t) be a smooth function in ξ rapidly decreas-ing as |ξ | → +∞. The Fourier transform φˆ(ξ ,z, t) of φ(y,z, t) in the y-variable,which is the distribution defined by〈φˆ(·,z, t),µ〉Rn−1 = 〈φ(·,z, t), µˆ〉Rn−1 =∫Rn−1φ(ξ ,z, t)µˆ(ξ )dξfor any smooth rapidly decreasing function µ , satisfies∫Rn+1+(−∇ · (ta∇ψ)+ ta|ξ |2ψ)φˆ(ξ ,z, t)dξdzdt=∫Rn(− f ′(w)ψ+ taψt |t=0) φˆ(ξ ,z,0)dξdz.Let µ ∈C∞c (Rn−1), ϕ+ ∈C∞c (R2+) and ϕ0 ∈C∞c (R) such that0 /∈ supp(µ).By Lemma 2.4.2, for any ξ 6= 0 we can solve the equation−∇ · (ta∇ψ)+ ta|ξ |2ψ = µ(ξ )ϕ+(z, t) in R2+ta∂ψ∂ν+ f ′(w)ψ = µ(ξ )ϕ0(z) on ∂R2+uniquely for ψ(ξ , ·, ·) ∈ X such thatψ(ξ ,z, t) = 0 if ξ /∈ supp(µ).37In particular, ψ(·,z, t) is rapidly decreasing for any (z, t) ∈ R2+. This implies∫R2+〈φˆ(·,z, t),µ〉Rn−1ϕ+(z, t)dzdt =∫R〈φˆ(·,z,0),µ〉Rn−1ϕ0(z)dzfor any ϕ+ ∈C∞c (R2+) and ϕ0 ∈C∞c (R). In other words, whenever 0 /∈ supp(µ), wehave〈φˆ(·,z, t),µ〉Rn−1 = 0 for all (z, t) ∈ R2+.Such distribution with supp(φˆ(·,z, t)) ⊂ {0} is characterized as a linear combina-tion of derivatives up to a finite order of Dirac masses at zero, namelyφˆ(ξ ,z, t) =N∑j=0a j(z, t)δ( j)0 (ξ ),for some integer N ≥ 0. Taking inverse Fourier transform, we see that φ(y,z, t) is apolynomial in y with coefficients depending on (z, t). Since we assumed that φ isbounded, it is a zeroth order polynomial, i.e. φ is independent of y. Now the traceφ(z,0) solves(−∆)sφ + f ′(w)φ = 0 in R.By Lemma 2.4.3,φ(z, t) =Cwz(z, t)for some constant C ∈ R. This completes the proof.2.4.2 A priori estimatesConsider the equation(−∆)sφ(y,z)+ f ′(w(z))φ(y,z) = g(y,z) for (y,z) ∈ Rn. (2.20)Let 〈y〉=√1+ |y|2 and define the norm‖φ‖µ,σ = sup(y,z)∈Rn〈y〉µ 〈z〉σ |φ(y,z)|for 0≤ µ < n−1+2s and 2−2s < σ < 1+2s such that µ+σ < n+2s.38Lemma 2.4.5 (Decay in z). Let φ ∈ L∞(Rn) and ‖g‖0,σ <+∞. Then we have‖φ‖0,σ ≤C.With the decay established, the following orthogonality condition (2.21) iswell-defined.Lemma 2.4.6 (A priori estimate in y,z). Let φ ∈ L∞(Rn) and ‖g‖µ,σ <+∞. If thes-harmonic extension φ(t,y,z) is orthogonal to wz(t,z) in Rn+1+ , namely,∫∫R2+taφwz dtdz = 0, (2.21)then we have‖φ‖µ,σ ≤C‖g‖µ,σ .Before we give the proof, we estimate some integrals which arise from theproduct rule(−∆)s(uv)(x0) = u(x0)(−∆)sv(x0)+Cn,s∫Rnu(x0)−u(x)|x0− x|n+2sv(x)dx= u(x0)(−∆)sv(x0)+ v(x0)(−∆)su(x0)− (u,v)s(x0),where(u,v)s(x0) =Cn,s∫Rn(u(x0)−u(x))(v(x0)− v(x))|x0− x|n+2sdx.Lemma 2.4.7 (Decay estimates). Suppose φ(y,z) is a bounded function.1. As |y| →+∞,(−∆)s 〈y〉−µ = O(〈y〉−2s−min{µ,n−1}),(φ ,〈y〉−µ)s = O(〈y〉−2s−min{µ,n−1}).2. As |z| →+∞,(−∆)s 〈z〉−σ = O(〈z〉−2s−min{σ ,1}),(φ ,〈z〉−σ )s = O(〈z〉−2s−min{σ ,1}).393. As min{|y|, |z|} →+∞,(〈y〉−µ ,〈z〉−σ )s = O(|(y,z)|−n−2s(|y|n−1−µ +1)(|z|1−σ +1))+O(|y|−n−2s(|y|n−1−µ +1)|z|−σ−2 min{|y|, |z|}3)+O(|y|−µ−2|z|−n−2s(|z|1−σ +1)min{|y|, |z|}n+1)+O(|z|−σ (|y|+ |z|)−(n−1+2s) (|y|n−1−µ +1))+O(|y|−µ (|y|+ |z|)−1−2s (|z|1−σ +1))+O(|y|−µ |z|−σ (|y|+ |z|)−2s).In particular, if µ < n−1+2s and σ < 1+2s, then(〈y〉−µ ,〈z〉−σ )s = o(|y|−µ |z|−σ) as min{|y|, |z|} →+∞.4. Suppose µ < n−1+2s and σ < 1+2s. As min{|y|, |z|} →+∞,(−∆)s (〈y〉−µ 〈z〉−σ)= o(|y|−µ |z|−σ) ,(φ ,〈y〉−µ 〈z〉−σ )s = o(|y|−µ |z|−σ) .5. Suppose ηR(y) = η( |y|R)where η is a smooth cut-off function as in (2.25),and φ(y,z)≤C 〈z〉−σ . For all sufficiently large R> 0, we have|[(−∆)s,ηR]φ(y,z)| ≤C(〈z〉−1+ 〈z〉−σ)max{|y|,R}−2s . (2.22)Let us assume the validity of Lemma 2.4.7 for the moment.Proof of Lemma 2.4.5. It follows from Lemma 2.4.7(2) and a maximum principle[50].Proof of Lemma 2.4.6. We will first establish the a priori estimate assuming that‖φ‖µ,σ < +∞. We use a blow-up argument. Suppose on the contrary that there40exist a sequence φm(y,z) and hm(y,z) such that(−∆)sφm+ f ′(w)φm = gm for (y,z) ∈ Rnand‖φm‖µ,σ = 1 and ‖gm‖µ,σ → 0 as m→+∞.Then there exist a sequence of points (ym,zm) ∈ Rn such thatφm(ym,zm)〈ym〉µ 〈zm〉σ ≥ 12 . (2.23)We consider four cases.1. ym, zm bounded:Since φm is bounded and gm→ 0 in L∞(Rn), by elliptic estimates and passingto a subsequence, we may assume that φm converges uniformly in compactsubsets of Rn to a function φ0 which satisfies(−∆)sφ0+ f ′(w)φ0 = 0, in Rnand, by (2.21), ∫∫R2+taφ0wz dtdz = 0.By the non-degeneracy of w′ (Lemma 2.4.4), we necessarily have φ0(y,z) =Cw′(z). However, the orthogonality condition yields C = 0, i.e. φ0 ≡ 0. Thiscontradicts (2.23).2. ym bounded, |zm| → ∞:We consider φ˜m(y,z) = 〈zm+ z〉σ φm(y,zm+ z), which satisfies in Rn〈zm+ z〉−σ (−∆)sφ˜m(y,z)+ φ˜m(y,z)(−∆)s 〈zm+ z〉−σ− (φ˜m(y,z),〈zm+ z〉σ)s+ f ′(w(zm+ z))〈zm+ z〉−σ φ˜m(y,z) = gm(y,zm+ z),41or(−∆)sφ˜m+(f ′(w(zm+ z))+(−∆)s 〈zm+ z〉−σ〈zm+ z〉−σ)φ˜m= gm+(φ˜m(y,z),〈zm+ z〉σ)s〈zm+ z〉−σ.Using Lemma 2.4.7(2), the limiting equation is(−∆)sφ˜0+2φ˜0 = 0 in Rn.Thus φ˜0 = 0, contradicting (2.23).3. |ym| → ∞, zm bounded:We define φ˜m(y,z) = 〈ym+ y〉µ φm(ym+ y,z), which satisfies(−∆)sφ˜m(y,z)+(f ′(w(z))+(−∆)s (〈ym+ y〉−µ)〈ym+ y〉−µ)φ˜m(y,z)= gm(ym+ y,z)+(φ˜m(y,z),〈ym+ y〉−µ)s〈ym+ y〉−µin Rn.By Lemma 2.4.7(1), the subsequential limit φ˜0 satisfies(−∆)sφ˜0+ f ′(w)φ˜0 = 0 in Rn.This leads to a contradiction as in case (1).4. |ym|, |zm| → ∞:This is similar to case (2). In fact for φ˜m(y,z) = 〈ym+ y〉µ 〈zm+ z〉σ φm(ym+42y,zm+ z), we have(−∆)sφ˜m(y,z)+(f ′(w(zm+ z))+(−∆)s (〈ym+ y〉−µ 〈zm+ z〉−σ)〈ym+ y〉−µ 〈zm+ z〉−σ)φ˜m(y,z)= gm(ym+ y,zm+ z)+(φ˜m(y,z),〈ym+ y〉−µ 〈zm+ z〉σ)s〈ym+ y〉−µ 〈zm+ z〉−σin Rn.In the limiting situation φ˜m→ φ˜0, by Lemma 2.4.7(4),(−∆)sφ˜0+2φ˜0 = 0 in Rn,forcing φ˜0 = 0 which contradicts (2.23).We conclude that‖φ‖µ,σ ≤C‖g‖µ,σ provided ‖φ‖µ,σ <+∞. (2.24)Now we will remove the condition ‖φ‖µ,σ <+∞. By Lemma 2.4.5, we knowthat ‖φ‖0,σ <+∞. Let η : [0,+∞)→ [0,1] be a smooth cut-off function such thatη = 1 on [0,1] and η = 0 on [2,+∞). (2.25)Write ηR(y) = η( |y|R). We apply the above derived a priori estimate to ψ(y,z) =ηR(y)φ(y,z), which satisfies(−∆)sψ+ f ′(w)ψ = ηRg+φ(−∆)sηR− (ηR,φ)s. (2.26)It is clear that ‖ηRg‖µ,σ ≤ ‖g‖µ,σ and ‖φ(−∆)sηR‖µ,σ ≤ CR−2s because of theestimate (−∆)sη(|y|)≤C 〈y〉−(n−1+2s). By Lemma 2.4.7(5),|[(−∆)s,ηR]φ(y0,z0)| ≤C(|z0|−1+ |z0|−σ)max{|y0|,R}−2s .43For σ < 1 and 0≤ µ < 2s, this yields‖[(−∆)s,ηR]φ‖µ,σ ≤CR−(2s−µ).Therefore, (2.24) and (2.26) give‖ηRφ‖µ,σ ≤C‖g‖µ,σ +CR−2s+CR−(2s−µ).Letting R→+∞, we arrive at‖φ‖µ,σ ≤C‖g‖µ,σ ,as desired.Proof of Lemma 2.4.7. We will only prove the statements regarding the fractionalLaplacian of the explicit function. The associated assertion concerning the innerproduct with φ will follow from the same proof using its boundedness, since all theterms are estimated in absolute value.1. We have(−∆(y,z))s(〈y〉−µ)|y=y0 = (−∆y)s 〈y〉µ |y=y0=Cn−1,s∫Rn−1〈y0〉−µ −〈y〉−µ|y0− y|n−1+2sdy≡ I1+ I2+ I3+ I4,44whereI1 =Cn−1,s∫B |y0|2(y0)〈y0〉−µ −〈y〉−µ −D〈y〉−µ |y=y0(y0− y)|y0− y|n−1+2sdy,I2 =Cn−1,s∫B1(0)〈y0〉−µ −〈y〉−µ|y0− y|n−1+2sdy,I3 =Cn−1,s∫B |y0|2(0)\B1(0)〈y0〉−µ −〈y〉−µ|y0− y|n−1+2sdy,I4 =Cn−1,s∫Rn−1\(B |y0|2(y0)∪B |y0|2(0)) 〈y0〉−µ −〈y〉−µ|y0− y|n−1+2sdy.45If |y0| ≤ 1, it is simple to get boundedness since 〈y〉−µ is smooth andbounded. For |y0| ≥ 1, we compute|I1|.∫B |y0|2(y0)∣∣D2 〈y〉−µ |y=y0 [y0− y]2∣∣|y0− y|n−1+2sdy. |y0|−µ−2∫ |y0|20ρ2ρ1+2sdρ. |y0|−(µ+2s),|I2|.∫B1(0)1|y0|n−1+2sdy. |y0|−(n−1+2s),|I3|. |y0|−(n−1+2s)∫B |y0|2(0)\B1(0)(〈y0〉−µ + |y|−µ) dy. |y0|−(n−1+2s)∫ |y0|21(〈y0〉−µ +ρ−µ)ρn−2 dρ. |y0|−(n−1+2s)(〈y0〉−µ (|y0|n−1−1)+ |y0|−µ+n−1−1). |y0|−(µ+2s)+ |y0|−(n−1+2s),|I4|. |y0|−µ∫Rn−1\(B |y0|2(y0)∪B |y0|2(0)) 1|y0− y|n−1+2sdy. |y0|−µ∫ ∞|y0|21ρ1+2sdρ. |y0|−(µ+2s).2. This follows from the same proof as (1).3. We divideRn−1×R into 14 regions in terms of the relative size of |y|, |z|withrespect to |y0|, |z0| which tend to infinity. We will consider such distance“small” if |y| < 1 and “intermediate” if 1 < |y| < |y0|2 , similarly for z. Oncethe non-decaying part of 〈y〉−µ ,〈z〉−σ are excluded, the remaining parts canbe either treated radially where we consider (y0,z0) as the origin, or reduced46to the one-dimensional case. More precisely, we write(〈y〉−µ ,〈z〉−σ )s(y0,z0) =Cn,s∫∫Rn(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz≡ ∑1≤i, j≤4min{i, j}≤2Ii j + Ising+ Irest ,whereI11 =Cn,s∫∫|y|<1, |z|<1(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I12 =Cn,s∫∫|y|<1,1<|z|< |z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I13 =Cn,s∫∫|y|<1, |z−z0|< |z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I14 =Cn,s∫∫|y|<1,min{|z|,|z−z0|}> |z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I21 =Cn,s∫∫1<|y|< |y0|2 , |z|<1(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I22 =Cn,s∫∫1<|y|< |y0|2 ,1<|z|<|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I23 =Cn,s∫∫1<|y|< |y0|2 , |z−z0|<|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I24 =Cn,s∫∫1<|y|< |y0|2 ,min{|z|,|z−z0|}>|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,47I31 =Cn,s∫∫|y−y0|< |y0|2 , |z|<1(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I32 =Cn,s∫∫|y−y0|< |y0|2 ,1<|z|<|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I41 =Cn,s∫∫min{|y|,|y−y0|}> |y0|2 , |z|<1(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,I42 =Cn,s∫∫min{|y|,|y−y0|}> |y0|2 ,1<|z|<|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,Ising =Cn,s∫∫|y|> |y0|2 , |z|>|z0|2 , |(y−y0,z−z0)|<|y0|+|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,Irest =Cn,s∫∫|y|> |y0|2 , |z|>|z0|2 , |(y−y0,z−z0)|>|y0|+|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz.We will estimate these integrals one by one. In the unit cylinder we have|I11|. 1|(y0,z0)|n+2s∫∫|y|<1, |z|<1dydz. |(y0,z0)|−n−2s.On a thin strip near the origin,|I12|. 1|(y0,z0)|n+2s∫∫|y|<1,1<|z|< |z0|2(|z|−σ + 〈z0〉−σ) dydz. |(y0,z0)|−n−2s(|z0|1−σ +1).Similarly|I21|. 1|(y0,z0)|n+2s∫∫1<|y|< |y0|2 , |z|<1(|y|−µ + 〈y0〉−µ) dydz. |(y0,z0)|−n−2s(|y0|n−1−µ +1),48and in the intermediate rectangle,|I22|.∫∫1<|y|< |y0|2 ,1<|z|<|z0|2(|y|−µ + 〈y0〉−µ)(|z|−σ + 〈z0〉−σ) dydz. |(y0,z0)|−n−2s(|y0|n−1−µ +1)(|z0|1−σ +1).The integral on a thin strip afar is more involved. We first integrate the zvariable by a change of variable z = z0+ |y0− y|ζ .I13 =Cn,s∫∫|y|<1, |z−z0|< |z0|2(〈y〉−µ −〈y0〉−µ)|(y− y0,z− z0)|n+2s(〈z〉−σ −〈z0〉−σ −D〈z〉−σ |z0(z− z0)) dydz=Cn,s∫∫|y|<1, |z−z0|< |z0|2(〈y〉−µ −〈y0〉−µ)|(y− y0,z− z0)|n+2s(z− z0)2(∫ 10(1− t)D2 〈z〉−σ |z0+t(z−z0) dt)dydz=Cn,s∫|y|<1〈y〉−µ −〈y0〉−µ|y− y0|n−3+2s∫|ζ |< |z0|2|y−y0|(∫ 10(1− t)D2 〈z〉−σ |z0+t|y−y0|ζ dt)ζ 2 dζ(1+ζ 2) n+2s2dy.Observing that in this regime |y− y0| ∼ |y0| and that∫ T0t2(1+ t2)n+2s2dt .min{T 3,1},we have|I13|.∫|y|<11|y− y0|n−3+2s|z0|−σ−2 min{( |z0||y− y0|)3,1}dy. |y0|−n−2s|z0|−σ−2 min{|y0|, |z0|}3 .49Similarly, changing y = y0+ |z− z0|η , we haveI31 =Cn,s∫∫|y−y0|< |y0|2 , |z|<1(〈y〉−µ −〈y0〉−µ −D〈y〉−µ |y0 · (y− y0))|(y− y0,z− z0)|n+2s(〈z〉−σ −〈z0〉−σ) dydz=Cn,s∫∫|y−y0|< |y0|2 , |z|<1(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2s·(n−1∑i, j=1∫ 10(1− t)∂i j 〈y〉−µ |y0+t(y−y0) dt)(y− y0)i(y− y0) j dydz=n−1∑i, j=1∫|z|<1〈z〉−σ −〈z0〉−σ|z− z0|2s−1∫|η |< |y0|2|z−z0|(∫ 10(1− t)∂i j 〈y〉−µ |y0+t|z−z0|η dt)ηiη j dη|(η ,1)|n+2s dz.The t-integral is controlled by 〈y0〉−µ−2 since∣∣y0 + t|z− z0|η∣∣ < |y0|2 . Thenusing∫|η |<η0|ηi|∣∣η j∣∣(|η |2+1) n+2s2dη .∫ η00ρ2ρn−2(ρ2+1) n+2s2dρ.min{ηn+10 ,1},(noting that here we again require s > 1/2) we have|I31|.n−1∑i, j=1∫|z|<11|z− z0|2s−1〈y0〉−µ−2 min{( |y0||z− z0|)n+1,1}dz. |z0|−n−2s 〈y0〉−µ−2 min{|y0|, |z0|}n+1 .Next we deal with the y-intermediate, z-far regions, namely I23. The treat-ment is similar to that of I13 except that we need to integrate in y. We have,50as above,I23 =Cn,s∫∫1<|y|< |y0|2 , |z−z0|<|z0|2(〈y〉−µ −〈y0〉−µ)|(y− y0,z− z0)|n+2s(〈z〉−σ −〈z0〉−σ −D〈z〉−σ |z0(z− z0)) dydz=Cn,s∫1<|y|< |y0|2〈y〉−µ −〈y0〉−µ|y− y0|n−3+2s∫|ζ |< |z0|2|y−y0|(∫ 10(1− t)D2 〈z〉−σ |z0+t|y−y0|ζ dt)ζ 2 dζ(1+ζ 2) n+2s2dy.Hence|I23|.∫1<|y|< |y0|2|y|−µ + 〈y0〉−µ|y− y0|n−3+2s|z0|−σ−2 min{( |z0||y− y0|)3,1}dy. |y0|−n−2s|z0|−σ−2 min{|y0|, |z0|}3∫1<|y|< |y0|2(|y|−µ + 〈y0〉−µ) dy. |y0|−n−2s|z0|−σ−2 min{|y0|, |z0|}3(|y0|n−1−µ +1).Similarly, we estimateI32 =Cn,s∫∫|y−y0|< |y0|2 ,1<|z|<|z0|2(〈y〉−µ −〈y0〉−µ −D〈y〉−µ |y0 · (y− y0))|(y− y0,z− z0)|n+2s(〈z〉−σ −〈z0〉−σ) dydz=n−1∑i, j=1∫1<|z|< |z0|2〈z〉−σ −〈z0〉−σ|z− z0|2s−1∫|η |< |y0|2|z−z0|(∫ 10(1− t)∂i j 〈y〉−µ |y0+t|z−z0|η dt)ηiη j dη|(η ,1)|n+2s dz,51which yields|I32|.n−1∑i, j=1∫1<|z|< |z0|2|z|−σ + 〈z0〉−σ|z− z0|2s−1〈y0〉−µ−2 min{( |y0||z− z0|)n+1,1}dz. |z0|−n−2s|y0|−µ−2 min{|y0|, |z0|}n+1∫1<|z|< |z0|2(|z|−σ + 〈z0〉−σ) dz. |z0|−n−2s|y0|−µ−2 min{|y0|, |z0|}n+1(|z0|1−σ +1).We consider the remaining part of the small strip, namely I14 and I41. Usingthe change of variable z = z0+ |y0|ζ , we haveI14 =Cn,s∫∫|y|<1,min{|z|,|z−z0|}> |z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,|I14|. 〈z0〉−σ∫∫|y|<1,min{|z|,|z−z0|}> |z0|21|(y0,z− z0)|n+2sdydz. 〈z0〉−σ∫min{|z|,|z−z0|}> |z0|21|(y0,z− z0)|n+2sdz. 〈z0〉−σ 1|y0|n−1+2s∫|ζ |> |z0|2|y0| ,∣∣∣∣ζ− z0|y0|∣∣∣∣> |z0|2|y0|1|(1,ζ )|n+2s dζ. 〈z0〉−σ |y0|−(n−1+2s)∫ ∞|z0|2|y0|dζ(1+ζ 2) n+2s2. 〈z0〉−σ |y0|−(n−1+2s)min{1,( |z0||y0|)−(n−1+2s)}. 〈z0〉−σ min{|y0|−(n−1+2s), |z0|−(n−1+2s)}. 〈z0〉−σ (|y0|+ |z0|)−(n−1+2s) .Similarly, with y = y0+ |z0|η ,I41 =Cn,s∫∫min{|y|,|y−y0|}> |y0|2 , |z|<1(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,52|I41|. 〈y0〉−µ∫∫min{|y|,|y−y0|}> |y0|2 , |z|<11|(y− y0,z0)|n+2sdydz. 〈y0〉−µ |z0|−(1+2s)∫|η |> |y0|2|z0|dη(|η |2+1) n+2s2. 〈y0〉−µ |z0|−(1+2s)∫ ∞|y0|2|z0|ρn−2(ρ2+1) n+2s2dρ. 〈y0〉−µ |z0|−(1+2s)min{( |y0|2|z0|)−(1+2s),1}. 〈y0〉−µ (|y0|+ |z0|)−(1+2s) .In the remaining intermediate region, we first “integrate” in z by the changeof variable z = z0+ |y− y0|ζ as follows.I24 =Cn,s∫∫1<|y|< |y0|2 ,min{|z|,|z−z0|}>|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,|I24|. 〈z0〉−σ∫∫1<|y|< |y0|2 ,min{|z|,|z−z0|}>|z0|2|y|−µ + 〈y0〉−µ|(y− y0,z− z0)|n+2sdydz. 〈z0〉−σ∫1<|y|< |y0|2|y|−µ + 〈y0〉−µ|y− y0|n−1+2s∫|ζ |> |z0|2|y−y0| ,∣∣∣∣ζ− z0|y−y0|∣∣∣∣> |z0|2|y−y0|dζ(1+ζ 2) n+2s2dy. 〈z0〉−σ∫1<|y|< |y0|2|y|−µ + 〈y0〉−µ|y− y0|n−1+2smin{1,( |z0||y− y0|)−(n−1+2s)}dy. 〈z0〉−σ∫1<|y|< |y0|2(|y|−µ + 〈y0〉−µ)(|y− y0|+ |z0|)−(n−1+2s) dy. 〈z0〉−σ (|y0|+ |z0|)−(n−1+2s)∫1<|y|< |y0|2(|y|−µ + 〈y0〉−µ) dy. |y|n−1−µ 〈z0〉−σ (|y0|+ |z0|)−(n−1+2s) .53Similarly,I42 =Cn,s∫∫min{|y|,|y−y0|}> |y0|2 ,1<|z|<|z0|2(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz,|I42|. 〈y0〉−µ∫∫min{|y|,|y−y0|}> |y0|2 ,1<|z|<|z0|2|z|−σ + 〈z0〉−σ|(y− y0,z− z0)|n+2sdydz. 〈y0〉−µ∫1<|z|< |z0|2|z|−σ + 〈z0〉−σ|z− z0|1+2s∫|η |> |y0|2|z−z0|dη(|η |2+1) n+2s2dz. 〈y0〉−µ∫1<|z|< |z0|2|z|−σ + 〈z0〉−σ|z− z0|1+2smin{( |y0|2|z− z0|)−1−2s,1}dz. 〈y0〉−µ |z0|1−σ (|y0|+ |z0|)−(1+2s) .Now we estimate the singular part Ising. The only concern is that if, say,|y0| |z0|, then the line segment joining z0 and z may intersect the y-axis. Tofix the idea we suppose that |y0| ≥ |z0|. Having all estimates for the integralsin a neighborhood of the axes, one can factor out the decay 〈z〉−σ −〈z0〉−σand obtain integrability by expanding the bracket with y to second order, asfollows. For simplicity let us writeΩsing ={(y,z) ∈ Rn : |y|> |y0|2, |z|> |z0|2, |(y− y0,z− z0)|< |y0|+ |z0|2}.ThenIsing =Cn,s∫∫Ωsing(〈y〉−µ −〈y0〉−µ)(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2sdydz=Cn,s∫∫Ωsing(〈z〉−σ −〈z0〉−σ)|(y− y0,z− z0)|n+2s·(n−1∑i, j=1∫ 10(1− t)∂i j 〈y〉−µ |y0+t(y−y0) dt)(y− y0)i(y− y0) j dydz.54Thus∣∣Ising∣∣. 〈z0〉−σ 〈y0〉−µ−2 ∫∫Ωsing|y− y0|2|(y− y0,z− z0)|n+2sdydz. 〈z0〉−σ 〈y0〉−µ−2∫ |y0|+|z0|20ρ2ρ1+2sdρ. 〈y0〉−µ−2s 〈z0〉−σ .The same argument implies that if |z0| ≥ |y0| then∣∣Ising∣∣. 〈y0〉−µ 〈z0〉−σ−2s .Therefore, we have in general∣∣Ising∣∣. 〈y0〉−µ 〈z0〉−σ max{|y0|, |z0|}−2s. 〈y0〉−µ 〈z0〉−σ (|y0|+ |z0|)−2s .Finally, the remaining exterior integral is controlled by∣∣Irest∣∣. 〈y0〉−µ 〈z0〉−σ ∫∫|y|> |y0|2 , |z|> |z0|2 , |(y−y0,z−z0)|< |y0|+|z0|21|(y− y0,z− z0)|n+2sdydz. 〈y0〉−µ 〈z0〉−σ∫ ∞|y0|+|z0|2dρρ1+2s. 〈y0〉−µ 〈z0〉−σ (|y0|+ |z0|)−2s .4. This follows from the product rule(−∆)s (〈y〉−µ 〈z〉−σ)= 〈y〉−µ (−∆)s 〈z〉−σ + 〈z〉−σ (−∆)s 〈y〉−µ − (〈y〉−µ ,〈z〉−σ )s= 〈y〉−µ 〈z〉−σ(O(〈y〉−2s)+O(〈z〉−2s)+o(1)).555. The s-inner product is computed as follows. We may assume that 1≤ |z0| ≤R2 . When |y0| ≥ 3R,|[(−∆)s,ηR]φ(y0,z0)|≤C∫Rn|−ηR(y)| 〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C∫R∫|y|≤2R〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤CRn−1∫R〈z〉−σ(|y0|2+ |z0− z|2) n+2s2dz≤CRn−1(∫|z|≥ |z0|2〈z0〉−σ(|y0|2+ |z0− z|2) n+2s2dz+∫|z|≤ |z0|2〈z〉−σ(|y0|2+ |z0|2) n+2s2dz)≤CRn−1(|z0|−σ |y0|−(n−1+2s)+(1+ |z0|1−σ )|(y0,z0)|−n−2s)≤C(|z0|−σ |y0|−2s+(|z0|−1+ |z0|−σ )|(y0,z0)|−2s)≤C(|z0|−1+ |z0|−σ)|y0|−2s.56When |y0| ≤ R2 ,|[(−∆)s,ηR]φ(y0,z0)|≤C∫Rn(1−ηR(y))〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C∫R∫|y|≥R〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C∫R∫|y|≥ R2〈z〉−σ(|y|2+ |z0− z|2) n+2s2dydz≤C∫R〈z〉−σ|z0− z|1+2s∫|y˜|≥ R2|z0−z|dy˜(|y˜|2+1) n+2s2dz≤C∫R〈z〉−σ|z0− z|1+2smin{1,( |z0− z|R)1+2s}dz≤C(∫ z0+Rz0−R〈z〉−σ R−1−2s dz+∫|z0−z|>R〈z〉−σ|z0− z|1+2sdz)≤C(R−1−2s(1+R1−σ )+R−σR−2s)≤C(R−1−2s+R−σ−2s) .When R2 ≤ |y0| ≤ 3R, we have∂yiy jηR =1R2η ′′( yR) yiy j|y|2 +1R|y|η′( yR)(δi j− yiy j|y|2),57which implies that∥∥D2ηR∥∥L∞([y0,y]) ≤ CR−2 for |y0− y| ≤ y02 , where [y0,y]denotes the line segment joining y0 and y. Thus|[(−∆)s,ηR]φ(y0,z0)|≤C∫Rn∣∣ηR(y0)−ηR(y)+χ{|y−y0|<1}DηR(y0) · (y− y0)∣∣〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C( ∫Rn−1∫|z|≤ |z0|2∣∣ηR(y0)−ηR(y)+χ{|y−y0|<1}DηR(y0) · (y− y0)∣∣(|y0− y|2+ |z0|2) n+2s2〈z〉−σ dydz+∫Rn−1∫|z|≥ |z0|2∣∣ηR(y0)−ηR(y)+χ{|y−y0|<1}DηR(y0) · (y− y0)∣∣〈z0〉−σ(|y0− y|2+ |z0− z|2) n+2s2dydz)≤C((1+ |z0|1−σ )∫Rn−1∣∣ηR(y0)−ηR(y)+χ{|y−y0|<1}DηR(y0) · (y− y0)∣∣(|y0− y|2+ |z0|2) n+2s2dy+ |z0|σ∫Rn−1∣∣ηR(y0)−ηR(y)+χ{|y−y0|<1}DηR(y0) · (y− y0)∣∣|y0− y|n−1+2sdy)≤C(|z0|−1+ |z0|−σ)( ∫|y0−y|≥ y02dy|y0− y|n−1+2s+∫|y0−y|≤ y02∥∥D2ηR∥∥L∞([y0,y]) |y0− y|2|y0− y|n−1+2sdy)≤C(|z0|−1+ |z0|−σ)(|y0|−2s+R−2|y0|2−2s)≤C(|z0|−1+ |z0|−σ)|y0|−2s.This completes the proof of (2.22).582.4.3 ExistenceIn order to solve the linearized equation(−∆)sφ + f ′(w)φ = g for (y,z) ∈ Rn,we consider the equivalent problem in the Caffarelli–Slivestre extension [43],−∇ · (ta∇φ) = 0 for (t,y,z) ∈ Rn+1+ta∂φ∂ν+ f ′(w)φ = g for (y,z) ∈ ∂Rn+1+ .(2.27)We will prove the followingProposition 2.4.8. Let µ,σ > 0 be small. For any g with ‖g‖µ,σ <+∞ satisfying∫Rg(y,z)w′(z)dz = 0, (2.28)there exists a unique solution φ ∈ H1(Rn+1+ , ta) of (2.27) satisfying∫∫R2+taφ(t,y,z)wz(t,z)dtdz = 0 for all y ∈ Rn−1, (2.29)such that the trace φ(0,y,z) satisfies ‖φ‖µ,σ <+∞. Moreover,‖φ‖µ,σ ≤C‖g‖µ,σ . (2.30)Let us recall the corresponding known result [82] in one dimension.Lemma 2.4.9. Let n = 1. For any g with∫R gw′ dz = 0, there exists a uniquesolution φ to (2.27) satisfying∫∫R2+ taφwz dtdz = 0 such that‖φ‖0,σ ≤C‖g‖0,σ .Proof. This is Proposition 4.1 in [82]. In their notations, take m = 1, ξ1 = 0 andµ = σ .59Proof of Proposition 2.4.8. 1. We first assume that g∈C∞c (Rn). Taking Fouriertransform in y, we solve for each ξ ∈ Rn−1 a solution φˆ(t,ξ ,z) to−∇ · (ta∇φˆ)+ |ξ |2taφˆ = 0 for (t,z) ∈ R2+,ta∂ φˆ∂ν+ f ′(w)φˆ = gˆ for z ∈ ∂R2+,with orthogonality condition∫∫R2+taφˆ(t,ξ ,z)wz(t,z)dtdz = 0 for all ξ ∈ Rn−1corresponding to (2.29). One can then obtain a solution for ξ = 0 by Lemma2.4.9 and for ξ 6= 0 by Lemma 2.4.2. From the embedding H1(R2+, ta) ↪→Hs(R) [36], we have the estimate∥∥φˆ(·,ξ , ·)∥∥H1(R2+,ta) ≤C(ξ )‖gˆ(ξ , ·)‖L2(R) .We claim that the constant can be taken independent of ξ , i.e.∥∥φˆ(·,ξ , ·)∥∥H1(R2+,ta) ≤C‖gˆ(ξ , ·)‖L2(R) . (2.31)If this were not true, there would exist sequences ξm→ 0 (the case |ξm| →+∞ is similar), φˆm and gˆm such that∥∥φˆm(·,ξm, ·)∥∥H1(R2+,ta) = 1, ‖gˆm(ξm, ·)‖L2(R) = 0, (2.32)−∇ · (ta∇φˆm)+ |ξm|2taφˆm = 0 for (t,z) ∈ R2+,ta∂ φˆm∂ν+ f ′(w)φˆm = gˆm for z ∈ ∂R2+,and ∫∫R2+taφˆm(t,ξm,z)wz(t,z)dtdz = 0.60Elliptic regularity implies that a subsequence of φˆm(t,ξm,z) converges loc-ally uniformly in R2+ to some φˆ0(t,z), which solves weakly−∇ · (ta∇φˆ0) = 0 for (t,z) ∈ R2+ta∂ φˆ0∂ν+ f ′(w)φˆ0 = 0 for z ∈ ∂R2+.and ∫∫R2+taφˆ0(t,z)wz(t,z)dtdz = 0 for all ξ ∈ Rn−1.By Lemma 2.4.4, we conclude that φˆ0 = 0, contradicting (2.32). This proves(2.31).Integrating over ξ ∈Rn−1 and using Plancherel’s theorem, we obtain a solu-tion φ satisfying‖φ‖H1(Rn+1+ ,ta) ≤C‖g‖L2(Rn) .Higher regularity yields, in particular, φ ∈ L∞(Rn). Then (2.30) follows fromLemma 2.4.6.2. In the general case, we solve (2.27) with g replaced by gm ∈C∞c (Rn) whichconverges uniformly to g. Then the solution φm is controlled by‖φm‖µ,σ ≤C‖gm‖µ,σ ≤C‖g‖µ,σ .By passing to a subsequence, φm converges to some φ uniformly on compactsubsets of Rn, which also satisfies (2.30).3. The uniqueness follows from the non-degeneracy of w′ and the orthogonalitycondition (2.29).2.4.4 The positive operatorWe conclude this section by stating a standard estimate for the operator (−∆)s+2.61Lemma 2.4.10. Consider the equation(−∆)su+2u = g in Rn.and |g(x)| ≤C 〈x′〉−θ for all x ∈Rn and g(x) = 0 for x in Mε,R, a tubular neighbor-hood of Mε of width R. Then the unique solution u = ((−∆)s +2)−1g satisfies thedecay estimate|u(x)| ≤C〈x′〉−θ 〈dist(x,Mε,R)〉−2s .Proof. The decay in x′ follows from a maximum principle; that in the interfaceis seen from the Green’s function for (−∆)s + 2 which has a decay |x|−(n+2s) atinfinity [62].2.5 Fractional gluing system2.5.1 Preliminary estimatesWe have the followingLemma 2.5.1 (Some non-local estimates). For φ j ∈X j, j ∈J , the following holdstrue.1. (commutator at the near interface)∣∣[(−∆(y,z))s, η¯ ζ¯ ]φ¯i(y,z)∣∣≤C‖φi‖i,µ,σ 〈yi〉−θ Rn(R+ |(y,z)|)−n−2s.As a result,∑i∈I∣∣[(−∆(y,z))s,ζi]φi(x)∣∣≤Cr−θ supi∈I‖φi‖i,µ,σ(R+dist(x,supp ∑i∈Iζi))−2s.2. (commutator at the end)∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ+(y,z)∣∣≤C‖φ+‖+,µ,σ R−θ2 〈y〉−µ 〈z〉−1−2s ,62and similarly for φ−.3. (linearization at u∗)∑j∈J∣∣ζ j( f ′(w j)− f ′(u∗))φ j∣∣≤C supj∈J∥∥φ j∥∥ j,µ,σ(∑i∈IζiRµ+σ 〈yi〉−θ−4s2s+1 +(ζ++ζ−)R−θ2 〈y〉−µ).4. (change of coordinates around the near interface)∑i∈I∣∣((−∆x)s− (−∆(y,z))s)(ζiφi)(x)∣∣≤CRn+1+µ+σε ∥∥φ¯i∥∥i,µ,σ∑i∈Iζi 〈yi〉−θ + εθ〈dist(x,supp ∑i∈Iζi)〉−2s .5. (change of coordinates around the end)∣∣((−∆x)s− (−∆(y,z))s)(ζ+φ+)(x)∣∣≤Cr− 2(2s−τ)2s+1 ∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ 〈z〉−1−2s ,and similarly for φ−.In particular, all these terms are dominated by S(u∗).Proof of Lemma 2.5.1. 1. (a) Since φi ∈ Xi, we have for |(y0,z0)| ≥ 3R,∣∣[(−∆(y,z))s, η¯ ζ¯ ]φ¯i(y0,z0)∣∣≤C‖φi‖i,µ,σ∣∣∣∣∣∫|(y,z)|≤2R−η¯(y)ζ¯ (z)|(y0,z0)|n+2sRµ+σ 〈yi〉−θ 〈y〉−µ 〈z〉−σ dydz∣∣∣∣∣≤C‖φi‖i,µ,σ Rµ+σ 〈yi〉−θ |(y0,z0)|−n−2s∫|(y,z)|≤2R〈y〉−µ 〈z〉−σ dydz≤C‖φi‖i,µ,σ Rµ+σ (1+R1−σ )(1+Rn−1−µ)〈yi〉−θ |(y0,z0)|−n−2s≤CRn|(y0,z0)|−n−2s ‖φi‖i,µ,σ 〈yi〉−θ for σ < 1, µ < n−1.63(b) For R2 ≤ |(y0,z0)| ≤ 3R,∣∣[(−∆(y,z))s, η¯ ζ¯ ]φ¯i(y0,z0)∣∣≤C∫|y0−y|< R4∫|z0−z|< R4R−2(|y0− y|2+ |z0− z|2)(|y0− y|2+ |z0− z|2) n+2s2Rµ+σ ‖φi‖i,µ,σ 〈yi〉−θ 〈y〉−µ 〈z〉−σ dydz+C∫|y0−y|> R4∫|z0−z|> R41(|y0− y|2+ |z0− z|2) n+2s2Rµ+σ ‖φi‖i,µ,σ 〈yi〉−θ 〈y〉−µ 〈z〉−σ dydz≤CR−2s ‖φi‖i,µ,σ 〈yi〉−θ .(c) For 0≤ |(y0,z0)| ≤ R2 ,∣∣[(−∆(y,z))s, η¯ ζ¯ ]φ¯i(y0,z0)∣∣≤C‖φi‖i,µ,σ∫|(y,z)|≥R1− η¯(y)ζ¯ (z)|(y− y0,z− z0)|n+2sRµ+σ 〈yi〉−θ 〈y〉−µ 〈z〉−σ dydz≤CR−2s ‖φi‖i,µ,σ 〈yi〉−θ .2. We consider different cases according to the values of the cut-off functionsη¯+(y) and ζ¯ (z).64(a) When η¯+(y0)ζ¯ (z0) = 0 with |y0| ≥ 2R2 and |z0| ≥ 3R,∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ+(y0,z0)∣∣≤C∥∥φ¯+∥∥+,µ,σ R−θ2 ∫|y|>R2∫|z|<2R〈y〉−µ 〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C∥∥φ¯+∥∥+,µ,σ R−θ2 (1+R1−σ )∫|y|>R2 〈y〉−µ(|y0− y|2+ |z0|2) n+2s2dy≤C∥∥φ¯+∥∥+,µ,σ R−θ2 (1+R1−σ )(∫R2<|y|≤ |y0|2〈y〉−µ(|y0|2+ |z0|2) n+2s2dy+∫|y|≥ |y0|2〈y0〉−µ(|y0− y|2+ |z0|2) n+2s2dy)≤C∥∥φ¯+∥∥+,µ,σ R−θ2 (1+R1−σ )(|y0|n−1−µ|(y0,z0)|n+2s+〈y0〉−µ|z0|1+2s)≤C∥∥φ¯+∥∥+,µ,σ R−θ2 (1+R1−σ )〈y0〉−µ 〈z0〉−1−2s .(b) When η¯+(y0)ζ¯ (z0) = 0 with |y0| ≤ 2R2 and |z0| ≥ 3R,∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ+(y0,z0)∣∣≤C∥∥φ¯+∥∥+,µ,σ R−θ−µ2 (1+R1−σ )∫|y|>R2 dy(|y0− y|2+ |z0|2) n+2s2≤C∥∥φ¯+∥∥+,µ,σ R−θ−µ2 (1+R1−σ )|z0|−1−2s.(c) When η¯+(y0)ζ¯ (z0) = 0 with |y0| ≤ R2−2R,∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ+(y0,z0)∣∣≤C∥∥φ¯+∥∥+,µ,σ R−θ2 ∫|y|>R2∫|z|<2R〈y〉−µ 〈z〉−σ|(y0,z0)− (y,z)|n+2sdydz≤C∥∥φ¯+∥∥+,µ,σ R−θ−µ2 ∫|z|<2R 〈z〉−σ min{1|z0− z|1+2s,1R1+2s}dz≤C∥∥φ¯+∥∥+,µ,σ R−θ−µ2 (1+R1−σ )〈z0〉−1−2s .65(d) When 0≤ η¯+(y0)ζ¯ (z0)≤ 1 with |y0| ≥ R2−2R and 0≤ |z0| ≤ 3R,∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ+(y0,z0)∣∣≤C∫|y0−y|<R∫|z0−z|<RR−2(|y0− y|2+ |z0− z|2)(|y0− y|2+ |z0− z|2) n+2s2∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ 〈z〉−σ dydz+C∫|y0−y|>R∫|z0−z|>R1(|y0− y|2+ |z0− z|2) n+2s2∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ 〈z〉−σ dydz≤CR−2s∥∥φ¯+∥∥+,µ,σ R−θ2 〈y0〉−µ+C∥∥φ¯+∥∥+,µ,σ R−θ2 ∫|y0−y|>R 〈y〉−µ|y0− y|n−1+2sdy≤C∥∥φ¯+∥∥+,µ,σ R−θ2 |y0|−µ .3. For the localized inner terms,∑i∈I∣∣ζi( f ′(w)− f ′(u∗))φi∣∣≤C‖φi‖i,µ,σ ζiF2sε Rµ+σ 〈yi〉−θ≤C‖φi‖i,µ,σ ∑i∈IζiRµ+σ 〈yi〉−θ−4s2s+1 .The two terms at the ends are controlled by∣∣ζ±( f ′(w)− f ′(u∗))φ±∣∣≤C‖φ±‖±,µ,σ ζ±RσR−(θ−µ)2 〈y〉−µ .By summing up we obtain the desired estimate.664. By using Corollary 2.3.4 and (2.10), we have in the Fermi coordinates,∣∣((−∆x)s− (−∆(y,z))s)(ζiφi)(x)∣∣≤CRε∣∣(−∆(y,z))s(η¯ ζ¯ φ¯i)(y,z)∣∣+Cε2s∣∣(η¯ ζ¯ φ¯i)(y,z)∣∣≤CRε (η¯(y)ζ¯ (z)∣∣(−∆(y,z))sφ¯i(y,z)∣∣+ ∣∣[(−∆(y,z))s, η¯ ζ¯ ]φ¯i(y,z)∣∣)+Cε2s(η¯ ζ¯ φ¯i)(y,z)≤CRε(η¯(y)ζ¯ (z)Rµ+σ∥∥φ¯i∥∥i,µ,σ 〈yi〉−θ 〈y〉−µ 〈z〉−σ+∥∥φ¯i∥∥i,µ,σ 〈yi〉−θ Rn(R+ |(y,z)|)−n−2s)≤CRn+1+µ+σε ∥∥φ¯i∥∥i,µ,σ 〈yi〉−θ (η¯(y)ζ¯ (z)+(R+ |(y,z)|)−n−2s) .Going back to the x-coordinates and summing up over i ∈I , we have∑i∈I∣∣((−∆x)s− (−∆(y,z))s)(ζiφi)(x)∣∣≤CRn+1+µ+σε ∥∥φ¯i∥∥i,µ,σ·∑i∈Iζi 〈yi〉−θ + εθ〈dist(x,supp ∑i∈Iζi)〉−2s .5. Similarly, using Corollary 2.3.5 and (2.10),∣∣((−∆x)s− (−∆(y,z))s)(ζ+φ+)(x)∣∣≤Cr− 2(2s−τ)2s+1 ∣∣(−∆(y,z))s(η¯+ζ¯ φ¯+)(y,z)∣∣+Cr− 4sτ2s+1 ∣∣(η¯+ζ¯ φ¯+)(y,z)∣∣≤Cr− 2(2s−τ)2s+1 (η¯+(y)ζ¯ (z)∣∣(−∆(y,z))sφ¯+(y,z)∣∣+ ∣∣[(−∆(y,z))s, η¯+ζ¯ ]φ¯+(y,z)∣∣)+Cr−4sτ2s+1 (η¯+ζ¯ φ¯+)(y,z)≤Cr− 2(2s−τ)2s+1(η¯+(y)ζ¯ (z)∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ+∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ 〈z〉−1−2s)≤Cr− 2(2s−τ)2s+1 ∥∥φ¯+∥∥+,µ,σ R−θ2 〈y〉−µ 〈z〉−1−2s .672.5.2 The outer problem: Proof of Proposition 2.2.2We give a proof of Proposition 2.2.2 and solve φo in terms of (φ j) j∈J .Proof of Proposition 2.2.2. We solve it by a fixed point argument. By Corollary2.3.3 and Lemma 2.5.1, the right hand side go = go(φo) of (2.6) satisfies go = 0 inMε,R and‖go‖θ ≤Cεθ +‖η˜oN(ϕ)‖θ +∥∥η˜o(2− f ′(u∗))φo∥∥θ≤Cεθ +‖φo‖L∞(Rn) ‖φo‖θ +CR−2s ‖φo‖θ ,so that by Lemma 2.4.10,∥∥((−∆)s+2)−1go∥∥θ ≤ (C+C˜2εθ +C˜R−2s)εθ ≤ C˜εθ .Next we check that for φo,ψo ∈ Xo, go(φo)−go(ψo) = 0 in Mε,R as well as‖go(φo)−go(ψo)‖θ ≤∥∥∥∥∥N(φo+ ∑j∈Jζ jφ j)−N(ψo+ ∑j∈Jζ jφ j)∥∥∥∥∥θ+∥∥η˜o(2− f ′(u∗))(φo−ψo)∥∥θ≤C(εθ +R−2s)‖φo−ψo‖θ .Hence∥∥((−∆)s+2)−1 (go(φo)−go(ψo))∥∥θ ≤C(εθ +R−2s)‖φo−ψo‖θ .By contraction mapping principle, there is a unique solution φo = Φo((φ j) j∈J ).The Lipschitz continuity of Φo with respect to (φ j) j∈J can be obtained by takinga difference.2.5.3 The inner problem: Proof of Proposition 2.2.3Here we solve the inner problem for (φ j) j∈J , with the solution of the outer prob-lem φo =Φo((φ j) j∈J ) plugged in.68Proof of Proposition 2.2.3. Let us denote the right hand side of (2.10) by g j. Wenotice that the norms can can be estimated without the projection (up to a constant).Indeed, for any function h¯ with∥∥h¯∥∥µ,σ <+∞,∥∥∥∥(∫ 2R−2R ζ¯ (t)h¯(y, t)w′(t)dt)w′(z)∥∥∥∥µ,σ≤C∥∥h¯∥∥µ,σ supz∈R〈z〉−1−2s+σ≤C∥∥h¯∥∥µ,σ .Then, keeping in mind that a barred function denotes the corresponding one inFermi coordinates, we have‖η˜iS(u∗)‖i,µ,σ ≤ 〈yi〉θ sup|y|,|z|≤2R〈y〉µ 〈z〉σ · 〈yi〉−4s2s+1 〈z〉−(2s−1)≤CRµ 〈yi〉−(4s2s+1−θ)≤Cδ ,∥∥η˜i(2− f ′(u∗))Φo((φ j) j∈J )∥∥i,µ,σ≤ ∥∥η˜iΦo((φ j) j∈J )∥∥i,µ,σ≤ 〈yi〉θ sup|y|,|z|≤2R〈y〉µ 〈z〉σ ·∣∣∣Φo((φ j) j∈J )(y,z)∣∣∣≤ 〈yi〉θ sup|y|,|z|≤2R〈y〉µ 〈z〉σ · 〈yi〉−θ∥∥∥Φo((φ j) j∈J )∥∥∥θ≤CRµ+σεθ supj∈J∥∥φ j∥∥ j,µ,σ≤CRµ+σεθC˜δ ,69and ∥∥∥∥∥η˜iN(Φo((φ j) j∈J )+ ∑j∈Jζ jφ j)∥∥∥∥∥i,µ,σ≤C 〈yi〉θ sup|y|,|z|≤2R〈y〉µ 〈z〉σ∣∣∣∣∣∣∣∣Φo((φ j) j∈J )(y,z)+ ∑j∈Jsupp η˜i∩suppζ j 6= /0η¯ jζ¯ φ¯ j(y,z)∣∣∣∣∣∣∣∣2≤CRµ+σ 〈yi〉θ sup|y|,|z|≤2R〈yi〉−2θ(supj∈J∥∥φ j∥∥ j,µ,σ)2+ ∑j∈Jsupp η˜i∩suppζ j 6= /0〈y j〉−2θ ( supj∈J∥∥φ j∥∥ j,µ,σ)2≤CRµ+σ 〈yi〉−θ C˜δ supj∈J∥∥φ j∥∥ j,µ,σ≤CRµ+σεθC˜2δ 2.Using Lemma 2.5.1 and estimating as in the proof of Proposition 2.2.2, we havefor all i ∈I ,‖gi‖i,µ,σ ≤Cδ (1+Rµ+σεθC˜+Rµ+σεθC˜δ +o(1)).Now we estimate the functions φ± at the ends. We have similarly‖η˜+S(u∗)‖+,µ,σ ≤CRθ2 supy≥R2,z≤2R〈y〉µ 〈z〉σ 〈y〉− 4s2s+1 〈z〉−(2s−1)≤CR−(4s2s+1−µ−θ)2≤Cδ for R2 chosen large enough,70∥∥η˜+(2− f ′(u∗))Φo((φ j) j∈J )∥∥+,µ,σ≤CRθ2 supy≥R2,z≤2R〈y〉µ 〈z〉σ∣∣∣Φo((φ j) j∈J )(y,z)∣∣∣≤CRσRθ2 supy≥R2,z≤2R〈y〉µ · 〈y〉−θ εθ supj∈J∥∥φ j∥∥ j,µ,σ≤CRµ2 εθC˜δ (since µ ≤ θ)≤CC˜ε θ2 δ for µ chosen small enough,and ∥∥∥∥∥η˜+N(Φo((φ j) j∈J )+ ∑j∈Jζ jφ j)∥∥∥∥∥+,µ,σ≤CRθ2 supy≥R2,z≤2R〈y〉µ 〈z〉σ∣∣∣∣∣∣∣∣Φo((φ j) j∈J )(y,z)+ ∑j∈Jsupp η˜+∩suppζ j 6= /0η¯ jζ¯ φ¯ j(y,z)∣∣∣∣∣∣∣∣2≤CRσ supy≥R2,z≤2R〈y〉µ〈y〉−2θ(supj∈J∥∥φ j∥∥ j,µ,σ)2+ ∑j∈Jsupp η˜+∩suppζ j 6= /0〈y j〉−2θ η¯ j( supj∈J∥∥φ j∥∥ j,µ,σ)2≤CRσR−θ2 + ∑j∈Jsupp η˜+∩suppζ j 6= /0〈y j〉−θ(supj∈J∥∥φ j∥∥ j,µ,σ)2≤CRσεθC˜δ(supj∈J∥∥φ j∥∥ j,µ,σ)≤CRσεθC˜2δ 2.71Putting together these estimates together with the non-local terms yields, using thelinear theory (Proposition 2.4.8 and Lemma 2.4.6),supj∈J∥∥L−1g j∥∥ j,µ,σ ≤C supj∈J∥∥g j∥∥ j,µ,σ≤Cδ (1+o(1))≤ C˜δ .It suffices to check the Lipschitz continuity with respect to φ j ∈ X j. Supposeφ j,ψ j ∈ X j. Using (2.7), we have for instance〈yi〉θ sup|y|,|z|≤2R〈y〉µ 〈z〉σ∣∣∣Φo((φ j) j∈J )(y,z)−Φo((ψ j) j∈J )(y,z)∣∣∣+N(Φo((φ j) j∈J )+ ∑j∈Jζ jφ j)−N(Φo((ψ j) j∈J )+ ∑j∈Jζ jψ j)≤CRµ+σ sup|y|,|z|≤2R(1+δ )∥∥∥Φo((φ j) j∈J )(y,z)−Φo((ψ j) j∈J )(y,z)∥∥∥θ+δ 〈yi〉θ ∑j∈Jsupp η˜i∩suppζ j 6= /0η¯ jζ¯∣∣φ¯ j− ψ¯ j∣∣(y,z)≤CRµ+σδ supj∈J∥∥φ j−ψ j∥∥ j,µ,σ ,72andRθ2 sup|y|≥R2, |z|≤2R〈y〉µ 〈z〉σ∣∣∣Φo((φ j) j∈J )(y,z)−Φo((ψ j) j∈J )(y,z)∣∣∣+N(Φo((φ j) j∈J )+ ∑j∈Jζ jφ j)−N(Φo((ψ j) j∈J )+ ∑j∈Jζ jψ j)≤CRσRθ2sup|y|≥R2, |z|≤2R(1+δ )〈y〉µ−θ ∥∥∥Φo((φ j) j∈J )(y,z)−Φo((ψ j) j∈J )(y,z)∥∥∥θ+δ 〈y〉µ ∑j∈Jsupp η˜i∩suppζ j 6= /0η¯ jζ¯∣∣φ¯ j− ψ¯ j∣∣(y,z)≤CRσRµ2 δ supj∈J∥∥φ j−ψ j∥∥ j,µ,σ .Thereforesupj∈J∥∥L−1g j((φ j) j∈J )−L−1g j((ψ j) j∈J )∥∥ j,µ,σ ≤ o(1) supj∈J∥∥φ j−ψ j∥∥ j,µ,σand (φk)k∈J 7→ L−1g j((φk)k∈J ) defines a contraction mapping on the productspace endowed with the supremum norm for suitably chosen parameters R,R2 largeand ε,µ small. This concludes the proof.732.6 The reduced equation2.6.1 Form of the equation: Proof of Proposition 2.2.4Proof of Proposition 2.2.4. Recalling Proposition 2.2.1, in the near and intermedi-ate regions r ∈[1ε ,4R¯ε],ΠS(u∗)(r) = C¯HMε (r)+O(ε2s),whereC¯ =∫ 2R−2RcH(z)ζ (z)w′(z)dz.For the far region r ≥ 4R¯ε , let us assume that xn > 0 to fix the idea. Denote byΠ± the projections onto the kernels w′±(z) of the upper and lower leaves respect-ively, where w±(z) = w(z±). Then z− = −2Fε(r)(1+ o(1))− z+ and so from theasymptotic behavior w(z)∼z→+∞ 1− cwz2s , we haveΠ+3(w(z+)+w(z−))(1+w(z+))(1+w(z−))(r)=∫ 2R−2R3(w(z)+w(−2Fε(r)(1+o(1))− z))· (1+w(z))(1+w(−2Fε(r)(1+o(1))− z))ζ (z)w′(z)dz=− C¯±F2sε (r)(1+o(1)),whereC¯± =∫ 2R−2R3cw(1−w(z)2)ζ (z)w′(z)dz.Similarly this is also true for the projection onto w′−(z) with the same coefficientC¯±(r),Π−3(w(z+)+w(z−))(1+w(z+))(1+w(z−))(r) =− C¯±(r)F2sε (r)(1+o(1)).74The other projections are estimated as follows.Π+cH(z+)HMε (y+) =∫ 2R−2RcH(z)ζ (z)w′(z)dz ·HMε (y+) = C¯HMε (y+),Π+cH(z−)HMε (y−)(r) =∫ 2R−2RcH(2Fε(r)(1+o(1))− z)ζ (z)w′(z)dz ·HMε (y−)= O(F−(2s−1)ε ·F−2sε)= O(F−(4s−1)ε),Π−cH(z−)HMε (y−) = C¯HMε (y−),Π−cH(z+)HMε (y+) = O(F−(4s−1)ε).We conclude that for r ≥ 4R¯ε ,Π±S(u∗)(r) = C¯HMε (y)−C¯±(r)F2sε (r)(1+o(1)).Taking into account the quadratically small term and the solution of the outer prob-lem, the reduced equation readsC¯H[Fε ](r) = O(ε2s) for1ε≤ r ≤ 4R¯ε,C¯H[Fε ](r) =C¯±F2sε (r)(1+o(1)) for r ≥ 4R¯ε.By a scaling Fε(r) = ε−1F(εr), it suffices to solve1r rF ′(r)√1+F ′(r)2′ = O(ε2s−1) for 1≤ r ≤ 4R¯,1r rF ′(r)√1+F ′(r)2′ = C¯0ε2s−1F2s(r)(1+o(1)) for r ≥ 4R¯.75For large enough r one may approximate the mean curvature by ∆F = 1r (rF′)′.Hence, we arrive at1r rF ′(r)√1+F ′(r)2′ = O(ε2s−1) for 1≤ r ≤ 4R¯,F ′′(r)+F ′(r)r=C¯0ε2s−1F2s(r)(1+o(1)) for r ≥ 4R¯.Then the inverse G of F is introduced to deal with the singularity at r = 1 in theusual coordinates. Finally, the Lipschitz dependence of the error follows directlyfrom the previously involved computations.2.6.2 Initial approximationIn this section we study an ODE which is similar to the one in [63]. The reducedequation for Fε : [ε−1,+∞)→ [0,+∞) can be approximated byF ′′ε (r)+F ′ε(r)r=1F2sε (r), for all r large.Under the scaling Fε(r) = ε−1F(εr), the equation for F : [1,+∞)→ [0,+∞) isF ′′(r)+F ′(r)r=ε2s−1F2s(r), for all r large.For r small, we approximate F by the catenoid. More precisely, let fC(r) = log(r+√r2−1), r = |x′| ≥ 1, rε =( |logε|ε) 2s−12, and consider the Cauchy problemf ′′ε +f ′εr=ε2s−1f 2sεfor r > rε ,fε(rε) = fC (rε) =2s−12(|logε|+ log|logε|)+ log2+O(r−2ε ) ,f ′ε (rε) = fC (rε) = r−1ε(1+O(r−2ε)).Then an approximation F0 to F can be defined byF0(r) = fC(r)+χ (r− rε)( fε(r)− fC(r)), r ≥ 1,76where χ : R→ [0,1] is a smooth cut-off function withχ = 0 on (−∞,0] and χ = 1 on [1,+∞). (2.33)Note that f ′ε(r)≥ 0 for all r ≥ rε .Lemma 2.6.1 (Estimates near initial value). For rε ≤ r ≤ |logε|rε , we have12|logε| ≤ fε(r)≤C|logε|,f ′ε(r)≤Cr−1ε ,∣∣ f ′′ε (r)∣∣≤ 1r2 + C|logε|r2ε .In fact the last inequality holds for all r ≥ rε .Proof. It is more convenient to writefε(r) = |logε| f˜ε(r−1ε r)so that f˜ε satisfiesf˜ ′′ε +f˜ ′εr=1|logε| f˜ 2sε, for r > 1,f˜ε(1) =2s−12+2s−12log|logε||logε| +log2|logε| +O(ε2s−1|logε|2s),f˜ ′ε(1) =1|logε| +O(ε2s−1|logε|2s).To obtain a bound for the first derivative, we integrate once to obtainr f˜ ′ε(r)− f˜ ′ε(1) =1|logε|2∫ r1r˜f˜ε(r˜)2sdr˜ for r ≥ 1.77By the monotonicity of fε , hence f˜ε , we havef˜ ′ε(r)≤1r(f˜ ′ε(1)+12|logε|2 f˜ε(1)2sr2)≤ 1r|logε| +Cr|logε|2for r ≥ 1. In particular,f˜ ′ε(r)≤C|logε| for 1≤ r ≤ |logε|.This also impliesf˜ε(r)≤C for 1≤ r ≤ |logε|.From the equation we obtain an estimate for f˜ ′′ε by∣∣ f˜ ′′ε (r)∣∣≤ 1r f˜ ′ε(r)+ 1|logε|2 f˜ 2sε≤ 1r2|logε| +C|logε|2 ,for all r ≥ 1.To study the behavior of fε(r) near infinity, we writefε(r) = |logε|gε(r|logε|rε).Then gε(r) satisfiesg′′ε +g′εr=1g2sε, for r ≥ 1|logε| ,gε(1|logε|)=2s−12+2s−12log|logε||logε| +log2|logε| +O(ε2s−1|logε|2s),g′ε(1|logε|)= 1+O(ε2s−1|logε|2s).(2.34)78Lemma 2.6.2 (Long-term behavior). For any fixed δ0 > 0, there exists C > 0 suchthat for all r ≥ δ0, ∣∣∣gε(r)− r 22s+1 ∣∣∣≤Cr− 2s−12s+1 ,∣∣∣∣g′ε(r)− 22s+1r− 2s−12s+1∣∣∣∣≤Cr− 4s2s+1 ,∣∣g′′ε (r)∣∣≤Cr− 4s2s+1 .Proof. Consider the change of variable of Emden–Fowler type,gε(r) = r22s+1 h˜ε(t), t = logr ≥− log|logε|.Then h˜ε(t)> 0 solvesh˜′′ε +222s+1h˜′ε +(22s+1)2h˜ε =1h˜2sεfor t ≥− log|logε|.The function hε defined by h˜ε(t) =(2s+12) 22s+1 hε( 22s+1 t)satisfiesh′′ε +2h′ε +hε =1h2sεfor t ≥−2s+12log|logε|. (2.35)We will first prove a uniform bound for hε with its derivative using a Hamilto-nianGε(t) =12(h′ε)2+12(h2ε −1)+12s−1(1h(2s−1)ε−1),which satisfiesG′ε(t) =−2(h′ε)2 ≤ 0. (2.36)By Lemma 2.6.1, we havehε(0) = O(h˜ε(0)) = O(gε(1)) = O(1),h′ε(0) = O(h˜′ε(0)) = O(g′ε(1)−22s+1gε(1))= O(1).79Therefore, Gε(0) =O(1) as ε→ 0 and by (2.36), Gε(t)≤C for all t ≥ 0 and ε > 0small. This implies that for some uniform constant C1 > 0,0 <C−11 ≤ hε(t)≤C1 <+∞ and∣∣h′ε(t)∣∣≤C1, for all t ≥ 0. (2.37)In fact, (2.36) implies∫ t0h′ε(t˜)2 dt˜ = 2Gε(0)−2Gε(t)≤ 2Gε(0)≤C,with C independent of ε and t, hence∫ ∞0h′ε(t˜)2 dt˜ ≤C,uniform in small ε > 0. In particular, |h′ε(t)| → 0 as t → ∞. We claim that theconvergence is uniform and exponential. Indeed, let us define the HamiltonianG1,ε =12(h′′ε )2+12(h′ε)2(1+2sh2s+1ε)for the linearized equationh′′′ε +2h′′ε +(1+2sh2s+1ε)h′ε = 0.We haveG′1,ε =−2(h′′ε )2− s(2s+1)h′3εh2s+2ε.By the uniform bounds in (2.37), if we choose 2C2 = s(2s+ 1)C2s+31 + 1, thenG˜ε =C2Gε +G1,ε satisfiesG˜′ε ≤−(h′′ε )2− (h′ε)2.80Using (2.37) and the vanishing of the zeroth order term together with its derivativeat hε = 1, we haveG˜ε =C2(12(h′ε)2+12(h2ε −1)+12s−1(1h2s−1ε−1))+12(h′′ε )2+12(h′ε)2(1+2sh2s+1ε)≤C((h′′ε )2+(h′ε)2+(hε − 1h2sε)2)≤−CG˜′ε .It follows that for some constants C,δ0 > 0 independent of ε > 0 small,G˜ε(t)≤Ce−δ0t for all t ≥ 0and, in particular,|hε(t)−1|+∣∣h′ε(t)∣∣≤Ce− δ02 t , for all t ≥ 0.This implies that after a fixed t1 independent of ε , the point (hε(t1),h′ε(t1)) is suf-ficiently close to (1,0). Letv1 = hεv2 = h′ε +hε .Then (2.35) is equivalent to (v1v2)′=(−v1+ v2v−2s1 − v2). (2.38)For t1 large the point (v1(t1),v2(t1)) is sufficiently close to (1,1) which is a hyper-bolic equilibrium point of (2.38). Now the linearization of (2.38), namely(v1v2)′=(−1 1−2s −1)(v1−1v2−1),81has eigenvalues −1± i√2s. By applying a C1 conjugacy we obtain|(v1(t),v2(t))− (1,1)| ≤Ce−t for all t ≥ t1.This implies in turn|hε(t)−1|+∣∣h′ε(t)∣∣≤Ce−t for all t ≥ 0,∣∣h˜ε(t)−1∣∣+ ∣∣h˜′ε(t)∣∣≤Ce−t for all t ≥ 0,and for any fixed r0 > 0, there exists C > 0 such that for all r ≥ r0,∣∣∣gε(r)− r 22s+1 ∣∣∣≤Cr− 2s−12s+1 and ∣∣∣∣g′ε(r)− 22s+1r− 2s−12s+1∣∣∣∣≤Cr− 4s2s+1and, in view of (2.34), ∣∣g′′ε (r)∣∣≤Cr− 4s2s+1 .Corollary 2.6.3 (Properties of the initial approximation). We have the followingproperties of F0.• For 1≤ r ≤ rε , F0(r) = fC(r) = log(r+√r2−1) andF0(r) = log(2r)+O(r−2),F ′0(r) =1√r2−1 =1r+O(r−3),F ′′0 (r) =−1r2+O(r−4),F ′′′0 (r) =2r3+O(r−5).82• For rε ≤ r ≤ δ0|logε|rε where δ0 > 0 is fixed,12|logε| ≤ F0(r)≤C|logε|,F ′0(r)≤Cr−1ε ,∣∣F ′′0 (r)∣∣≤C( 1r2 + 1|logε|r2ε),∣∣F ′′′0 (r)∣∣≤Cr−1ε ( 1r2 + 1|logε|r2ε).• For r ≥ δ0|logε|rε , F0(r) = fε(r) andF0(r) = ε2s−12s+1 r22s+1 +O(ε−(2s−1)22(2s+1) |logε| 2s+12 r− 2s−12s+1),F ′0(r) =22s+1ε2s−12s+1 r−2s−12s+1 +O(ε−(2s−1)22(2s+1) |logε| 2s+12 r− 4s2s+1),F ′′0 (r) = O(ε2s−12s+1 r−4s2s+1),F ′′′0 (r) = O(ε2s−12s+1 r−6s+12s+1).Proof. They follow from Lemmata 2.6.1 and 2.6.2. For the third derivative, wedifferentiate the equation and use the estimates for the lower order derivatives.2.6.3 The linearizationNow we build a right inverse for the linearized operatorL0(φ)(r) = (1−χε(r))1r(rφ ′(1+F ′0(r)2)32)′+χε(r)(φ ′′+φ ′r+2sε2s−1F0(r)2s+1φ),where χε is any family of smooth cut-off functions with χε(r) = 0 for 1 ≤ r ≤ rεand χε(r) = 1 for r ≥ δ0|logε|rε where δ0 > 0 is a sufficiently small number. Thegoal is to solveL0(φ)(r) = h(r) for r ≥ 1, (2.39)in a weighted function space which allows the expected sub-linear growth. Let usrecall the norms ‖·‖∗ and ‖·‖∗∗ defined in (2.13) and (2.14).83Proposition 2.6.4. Let γ ≤ 2+ 2s−12s+1 . For all sufficiently small δ0,ε > 0, thereexists C > 0 such that for all h with ‖h‖∗∗ <+∞, there exists a solution φ = T (h)of (2.39) with ‖φ‖∗ <+∞ that defines a linear operator T of h such that‖φ‖∗ ≤C‖h‖∗∗and φ(1) = 0.We start with an estimate of the kernels of the linearized equation in the farregion, namelyZ′′+Z′r+2sε2s−1fε(r)2s+1Z = 0, for r ≥ δ0|logε|rε . (2.40)Lemma 2.6.5. There are two linearly independent solutions Z1, Z2 of (2.40) sothat for i = 1,2, we have|Zi(r)| ≤C(rrε |logε|)− 2s−12s+1and∣∣Z′i(r)∣∣≤ Crε |logε|(rrε |logε|)− 2s−12s+1for r ≥ δ0|logε|rε where δ0 > 0 is fixed and rε =( |logε|ε) 2s−12.Proof. We will show that the elements Z˜i of the kernel of the linearization aroundgε , which solveZ˜′′+Z˜′r+2sgε(r)2s+1Z˜ = 0 for r ≥ 1|logε| , (2.41)satisfies∣∣Z˜i(r)∣∣≤Cr− 2s−12s+1 and ∣∣Z˜′i(r)∣∣≤Cr− 2s−12s+1 for all r ≥ δ0for i = 1,2; the result then follows by setting Zi(r) = Z˜i(rrε |logε|).A first kernel Z˜1 can be obtained from the scaling invariance gε,λ (r) =λ−22s+1 gε(λ r) of (2.34), givingZ˜1(r) = rg′ε(r)−22s+1gε(r).84Then for Z˜2 we solve (2.41) with the initial conditionsZ˜2(δ0) =− Z˜′1(δ0)δ0(Z˜1(δ0)2+ Z˜′1(δ0)2) , Z˜′2(δ0) = Z˜1(δ0)δ0 (Z˜1(δ0)2+ Z˜′1(δ0)2)for a fixed δ0 > 0. In particular the Wron´skian W˜ = Z˜1Z˜′2− Z˜′1Z˜2 is computedexactly asW˜ (r) =δ0W˜ (δ0)r=1rfor all r >1|logε| . (2.42)As in the proof of Lemma 2.6.2, we write t = logr and consider the Emden–Fowler change of variable Z˜(r) = r22s+1 v˜(t) followed by a re-normalization v˜(t) =( 22s+1)− 22s+1 v( 22s+1 t) which yield respectivelyv˜′′+222s+1v˜′+((22s+1)2+2sh˜2s+1ε)v˜ = 0, for t ≥− log|logε|,v′′+2v′+(1+2s)v = 2s(1− 1h2s+1ε)v, for t ≥−2s+12log|logε|.From this point we may express v2(t), and hence Z˜2(r), as a perturbation of thelinear combination of the kernelse−t cos(√2st) and e−t sin(√2st).Now we show the existence of the right inverse.Proof of Proposition 2.6.4. We sketch the argument by obtaining a solution in aweighted L∞ space. The general case follows similarly.1. Note that we will need to control φ up to two derivatives in the intermediateregion. For this purpose, for any γ ∈ R and any interval I ⊆ [r1,+∞) wedefine the norm‖φ‖γ,I = supIrγ−2|φ(r)|+ supIrγ−1∣∣φ ′(r)∣∣+ supIrγ∣∣φ ′′(r)∣∣.85By solving the linearized mean curvature equation in the inner region usingthe variation of parameters formula, we obtain the estimate‖φ‖γ,[r1,rε ] ≤C‖rγh‖L∞([1,+∞)) ,which in particular gives a bound for φ together with its derivatives at rε .2. In the intermediate region we write the equation asφ ′′+φ ′r= h− h˜, rε ≤ r ≤ r˜ε ,wherer˜ε = δ0|logε|rε ,andh˜(r) = χε(r)2sε2s−1F ′0(r)2s+1φ(r)+(1−χε(r))((1− 1(1+F ′0(r)2)32)(φ ′′+φ ′r)+3F ′0(r)F′′0 (r)(1+F ′0(r)2)32φ ′)is small. Again we integrate to obtainφ(r) = φ(rε)+ rεφ ′(rε) logrrε+∫ rrε1t∫ trετ(h(t)− h˜(t))dτ dt,φ ′(r) =rεφ ′(rε)r+1r∫ rrεt(h(t)− h˜(t))dt,φ ′′(r) =−rεφ′(rε)r2+h(r)− h˜(r)− 1r2∫ rrεt(h(t)− h˜(t))dt.86Using Corollary 2.6.3 we have, for small enough δ0 and ε ,∥∥rγ h˜∥∥L∞([rε ,r˜ε ]) ≤C ε2s−1|logε|2s+1 r2 ‖φ‖γ,[rε ,r˜ε ]+C(ε|logε|)2s−1‖φ‖γ,[rε ,r˜ε ]+C(ε|logε|) 2s−12(1r2+ε2s−1|logε|2s)r‖φ‖γ,[rε ,r˜ε ]≤C(δ02+δ0(ε|logε|) 2s−12|logε|)‖φ‖γ,[rε ,r˜ε ]≤ δ0 ‖φ‖γ,[rε ,r˜ε ] .This implies‖φ‖γ,[rε ,r˜ε ] ≤C‖rγh‖L∞([1,+∞))+δ0 ‖φ‖γ,[rε ,r˜ε ] ,or‖φ‖γ,[rε ,r˜ε ] ≤C‖rγh‖L∞([1,+∞)) (2.43)which is the desired estimate.3. In the outer region, we need to solveφ ′′+φ ′r+2sε2s−1f 2s+1εφ = h, r > r˜ε .In terms of the kernels Zi given in Lemma 2.6.5, the Wron´skian W = Z1Z′2−Z′1Z2 is given byW (r) =1rε |logε|W˜(rrε |logε|)=1r(2.44)using (2.42). Using the variation of parameters formula, we may writeφ(r) = c1Z1(r)+ c2Z2(r)+φ0(r),87whereφ0(r) =−Z1(r)∫ rr˜ερZ2(ρ)h(ρ)dρ+Z2(r)∫ rr˜ερZ1(ρ)h(ρ)dρand the constants ci are determined byφ(r˜ε) = c1Z1(r˜ε)+ c2Z2(r˜ε),φ ′(r˜ε) = c1Z′1(r˜ε)+ c2Z′2(r˜ε).By Lemma 2.6.5, (2.44) and (2.43), we readily check that for i = 1,2,|φ0(r)| ≤C(rr˜ε)− 2s−12s+1 ∫ rr˜ερ(ρr˜ε)− 2s−12s+1ρ−γ ‖rγh‖L∞([1,+∞)) dρ≤Cr2−γ ‖rγh‖L∞([1,+∞)) ,|ci| ≤Cr1(Cr1r2−γ ‖rγh‖L∞([1,+∞))+Cr1−γ1 ‖rγh‖L∞([1,+∞)))≤Cr˜2−γε ‖rγh‖L∞([1,+∞)) ,|ci||Zi(r)| ≤C(rr˜ε)− 2s−12s+1−(2−γ)r2−γ ‖rγh‖L∞([1,+∞))≤Cr2−γ ‖rγh‖L∞([1,+∞)) since γ ≤ 2+2s−12s+1,from which we conclude∥∥rγ−2φ∥∥L∞([r˜ε ,+∞)) ≤C‖rγh‖L∞([1,+∞)) .2.6.4 The perturbation argument: Proof of Proposition 2.2.5We solve the reduced equationL (F) =N1[F ] for r ≥ 1, (2.45)88using the knowledge of the initial approximation F0 and the linearized operatorL0 at F0 obtained in Sections 2.6.2 and 2.6.3 respectively. We look for a solutionF = F0+φ . Then φ satisfiesL0φ = A[φ ] =N1[F0+φ ]−L (F0)−N2[φ ],where N2[φ ] = L (F0 + φ)−L (F0)−L ′(F0)φ and φ(0) = 0. In terms of theoperator T defined in Proposition 2.6.4, we can write it in the formφ = T (A[φ ]) . (2.46)We apply a standard argument using contraction mapping principle as in [63]. Firstwe note that the approximation L (F0) is small and compactly supported in theintermediate region. The non-linear terms in A[φ ] are also small in the norm ‖·‖∗∗.Hence T (A[φ ]) defines a contraction mapping in the space X∗. The details are leftto the interested readers.89Chapter 3Fractional Yamabe Problem3.1 IntroductionWe construct singular solutions to the following non-local semilinear problem(−∆Rn)γu = up in Rn, u > 0, (3.1)for γ ∈ (0,1), n≥ 2, where the fractional Laplacian is defined by(−∆Rn)γu(z) = kn,γP.V.∫Rnu(z)−u(z˜)|z− z˜|n+2γ dz˜, for kn,γ = pi−n/222γΓ(n2 + γ)Γ(1− γ) γ.(3.2)Equation (3.1) for the critical power p= n+2γn−2γ corresponds to the fractional Yamabeproblem in conformal geometry, which asks to find a constant fractional curvaturemetric in a given conformal class (see [86, 111, 112, 122, 129]). In particular, forγ = 1 the fractional curvature coincides with the scalar curvature modulo a mul-tiplicative constant, so (3.1) reduces to the classical Yamabe problem. However,classical methods for local equations do not generally work here and one needs todevelop new ideas.Non-local equations have attracted a great deal of interest in the communitysince they are of central importance in many fields, from the points of view ofboth pure analysis and applied modeling. By the substantial effort made in the pastdecade by many authors, we have learned that non-local elliptic equations do enjoy90good PDE properties such as uniqueness, regularity and maximum principle. How-ever, not so much is known when it comes to the study of an integro-differentialequation such as (3.1) from an ODE perspective since most of the ODE theoryrelies on local properties and phase-plane analysis; our first achievement is the de-velopment of a suitable theory for the fractional order ODE (3.6), that arises whenstudying radial singular solutions to (3.1).On the one hand, we construct singular radial solutions for (3.1) directly witha completely different argument. On the other hand, using ideas from conformalgeometry and scattering theory we replace phase-plane analysis by a global studyto obtain that solutions of the nonlocal ODE (3.6) do have a behavior similar inspirit to a classical second-order autonomous ODE, and initiate the study of a non-local phase portrait. In particular, we show that a linear non-local ODE has atwo-dimensional kernel. This is surprising since this non-local ODE has an infinitenumber of indicial roots at the origin and at infinity, which is very different from thelocal case where the solution to a homogeneous linear second order problem can bewritten as a linear combination of two particular solutions and thus, its asymptoticbehavior is governed by two pairs of indicial roots.Then, with these tools at hand, we arrive at our second accomplishment: todevelop a Mazzeo-Pacard gluing program [132] for the construction of singularsolutions to (3.1) in the non-local setting. This gluing method is indeed local bydefinition; so one needs to rethink the theory from a fresh perspective in orderto adapt it for such non-local equation. More precisely, the program relies onthe fact that the linearization to (3.1) has good properties. In the classical case,this linearization has been well studied applying microlocal analysis (see [130],for instance), and it reduces to the understanding of a second order ODE with tworegular singular points. In the fractional case this is obviously not possible. Instead,we use conformal geometry, complex analysis and some non-Euclidean harmonicanalysis coming from representation theory in order to provide a new proof.Thus conformal geometry is the central core in this chapter, but we provide aninterdisciplinary approach in order to approach the following analytical problem:Theorem 3.1.1. Let Σ =⋃Ki=1Σi be a disjoint union of smooth, compact sub-manifolds Σi without boundary of dimensions ki, i = 1, . . . ,K. Assume, in addition91to n− ki ≥ 2, thatn− kin− ki−2γ < p <n− ki+2γn− ki−2γ ,or equivalently,n− 2pγ+2γp−1 < ki < n−2pγp−1for all i = 1, . . . ,K. Then there exists a positive solution for the problem(−∆Rn)γu = up in Rn \Σ (3.3)that blows up exactly at Σ.As a consequence of the previous theorem we obtain:Corollary 3.1.2. Assume that the dimensions ki satisfy0 < ki <n−2γ2. (3.4)Then there exists a positive solution to the fractional Yamabe equation(−∆Rn)γu = un+2γn−2γ in Rn \Σ (3.5)that blows up exactly at Σ.The dimension estimate (3.4) is sharp in some sense. Indeed, it was proved byGonza´lez, Mazzeo and Sire [110] that, if such u blows up at a smooth sub-manifoldof dimension k and is polyharmonic, then k must satisfy the restrictionΓ(n4− k2+γ2)/Γ(n4− k2− γ2)> 0,which in particular, includes (3.4). Here, and for the rest of the chapter, Γ denotesthe Gamma function. In addition, the asymptotic behavior of solutions to (3.5)when the singular set has fractional capacity zero has been considered in [121].Let us describe our methods in detail. First, note that it is enough to let Σ be asingle sub-manifold of dimension k, and we will restrict to this case for the rest ofthe chapter. We denote N = n− k.92The first step is to construct the building block, i.e, a solution to (3.3) in Rn \Rk that blows up exactly at Rk. For this, we write Rn \Rk = (RN \ {0})×Rk,parameterized with coordinates z = (x,y), x ∈ RN \ {0}, y ∈ Rk, and construct asolution u1 that only depends on the radial variable r = |x|. Then u1 is also a radialsolution to(−∆RN )γu = AN,p,γup in RN \{0}, u > 0.We write u = r−2γp−1 v, r = e−t . Then, in the radially symmetric case, this equationcan be written as the integro-differential ODEP.V.∫RK(t− t ′)[v(t)− v(t ′)]dt ′+AN,p,γv(t) = AN,p,γvp in R, v > 0, (3.6)where the kernel K is given precisely in (3.80). However, in addition to having theright blow up rate at the origin, u1 must decay fast as r→∞ in order to perform theMazzeo-Pacard gluing argument later. The existence of such fast-decaying singularsolutions in the case of γ = 1 is an easy consequence of phase-plane analysis as(3.6) is reduced to a second order autonomous ODE (see Proposition 1 of [132]).The analogue in the fractional case turns out to be quite non-trivial. To show theexistence, we first use Kelvin transform to reduce our problem for entire solutionsto a supercritical one (3.13). Then we consider an auxiliary non-local problem(3.14), for which we show that the minimal solution wλ is unique using Schaaf’sargument as in [83] and a fractional Pohozˇaev identity [149]. A blow up argument,together with a Crandall-Rabinowitz bifurcation scheme yields the existence of thisu1. This is the content of Section 3.2.Then, in Section 3.3, we exploit the conformal properties of the equation toproduce a geometric interpretation for (3.3) in terms of scattering theory and con-formally covariant operators. Singular solutions for the standard fractional Lapla-cian in Rn \Rk can be better understood by considering the conformal metric gkfrom (3.45), that is the product of a sphere SN−1 and a half-space Hk+1. Inspiredby the arguments by DelaTorre and Gonza´lez [66], our point of view is to rewritethe well known extension problem in Rn+1+ for the fractional Laplacian in Rn dueto [43], as a different, but equivalent, extension problem and to consider the corres-ponding Dirichlet-to-Neumann operator Pgkγ , defined in SN−1×Hk+1. Here Rn+1+is replaced by anti-de Sitter (AdS) space, but the arguments run in parallel.93This Pgkγ turns out to be a conjugate operator for (−∆Rn)γ , (see (3.46)), andbehaves well when the nonlinearity in (3.3) is the conformal power. However,the problem (3.3) is not conformal for a general p, so we need to perform a furtherconjugation (3.57) and to consider the new operator P˜gkγ . Then the original equation(3.3) in Rn \Rk is equivalent toP˜gkγ (v) = vp in SN−1×Hk+1, v = r2γp−1 u, v > 0 and smooth. (3.7)Rather miraculously, both Pgkγ and P˜gkγ diagonalize under the spherical harmonicdecomposition of SN−1. In fact, they can be understood as pseudo-differentialoperators on hyperbolic space Hk+1, and we calculate their symbols in Theorem3.3.5 and Proposition 3.3.6, respectively, under the Fourier-Helgason transform(to be denoted by ·̂ ) on the hyperbolic space understood as the symmetric spaceM = G/K for G = SO(1,k+ 1) and K = SO(k+ 1) (see the Appendix for a shortintroduction to the subject). This is an original approach that yields new resultseven in the classical case γ = 1, simplifying some of the arguments in [132]. Theprecise knowledge of their symbols allows, as a consequence, for the developmentof the linear theory for our problem, as we will comment below.Section 3.4 collects these ideas in order to develop new methods for the studyof the non-local ODE (3.6), which is precisely the projection of equation (3.7) fork = 0, n = N, over the zero-eigenspace when projecting over spherical harmonicsof SN−1. The advantage of shifting from u to v is that we obtain a new equationthat behaves very similarly to a second order autonomous ODE. This includes theexistence of a Hamiltonian quantity along trajectories.Moreover, one can take the spherical harmonic decomposition of SN−1 andconsider all projections m= 0,1, . . .. In Proposition 3.4.2 we are able to write everyprojected equation as an integro-differential equation very similar to the m = 0projection (3.6). This formulation immediately yields regularity and maximumprinciples for the solution of (3.7) following the arguments in [65].Now, to continue with the proof of Theorem 3.1.1, one takes the fast decayingsolution in Rn \Rk we have just constructed and, after some rescaling by ε , glues itto the background Euclidean space in order to have a global approximate solutionu¯ε in Rn \Σ. Even though the fractional Laplacian is a non-local operator, one is94able to perform this gluing just by carefully estimating the tail terms that appear inthe integrals after localizaton. This is done in Section 3.5.1 and, more precisely,Lemma 3.5.7, where we show that the error we generate when approximating atrue solution by u¯ε , given byfε := (−∆Rn)γ u¯ε − u¯pε ,is indeed small in suitable weighted Ho¨lder spaces.Once we have an approximate solution, we define the linearized operatoraround it,Lεφ := (−∆Rn)γφ − pu¯p−1ε φ .The general scheme of Mazzeo-Pacard’s method is to set u = u¯ε + φ for an un-known perturbation φ and to rewrite equation (3.3) asLε(φ)+Qε(φ)+ fε = 0,where Qε contains the remaining nonlinear terms. If Lε is invertible, then we canwriteφ = (Lε)−1(−Qε(φ)− fε),and a standard fixed point argument for small ε will yield the existence of such φ ,thus completing the proof of Theorem 3.1.1 (see Section 3.9).Thus, a central argument here is the study of the linear theory for Lε and, inparticular, the analysis of its indicial roots, injectivity and Fredholm properties.However, while the behaviour of a second order ODE is governed by two boundaryconditions (or behavior at the singular points using Frobenius method), this maynot be true in general for a non-local operator.We first consider the model operatorL1 defined in (3.124) for an isolated sin-gularity at the origin. Near the singularityL1 behaves like(−∆RN )γ −κr2γ(3.8)or, after conjugation, like Pg0γ −κ , which is a fractional Laplacian operator with aHardy potential of critical type.95The central core of the linear theory deals with the operator (3.8). In Section3.6 we perform a delicate study of the Green’s function by inverting its Fouriersymbol Θmγ (see (3.38)). This requires a very careful analysis of the poles of thesymbol, in both the stable and unstable cases. Contrary to the local case γ = 1,in which there are only two indicial roots for each projection m, here we find aninfinite sequence for each m. But in any case, these are controlled. It is alsointeresting to observe that, even though we have a non-local operator, the firstpair of indicial roots governs the asymptotic behavior of the operator and thus, itskernel is two-dimensional in some sense (see, for instance, Proposition 3.6.10 fora precise statement).Then, in Section 3.7 we complete the calculation of the indicial roots (seeLemma 3.7.1). Next, we show the injectivity for L1 in weighted Ho¨lder spaces,and an a priori estimate (Lemma 3.7.4) yields the injectivity for Lε .In addition, in Section 3.8 we work with weighted Hilbert spaces and we proveFredholm properties for Lε in the spirit of the results by Mazzeo [130, 131] for edgetype operators by constructing a suitable parametrix with compact remainder. Thedifficulty lies precisely in the fact that we are working with a non-local operator,so the localization with a cut-off is the non-trivial step. However, by working withsuitable weighted spaces we are able to localize the problem near the singularity;indeed, the tail terms are small. Then we conclude that Lε must be surjective bypurely functional analysis reasoning. Finally, we construct a right inverse for Lε ,with norm uniformly bounded independently of ε , and this concludes the proof ofTheorem 3.1.1.The Appendix contains some well known results on special functions and theFourier-Helgason transform.As a byproduct of the proof of Theorem 3.1.1, we will obtain the existence ofsolutions with isolated singularities in the subcritical regime (note the shift from nto N in the spatial dimension, which will fit better our purposes).Theorem 3.1.3. Let γ ∈ (0,1), N ≥ 2 andNN−2γ < p <N+2γN−2γ . (3.9)96Let Σ be a finite number of points, Σ= {q1, . . . ,qK}. Then equation(−∆RN )γu = AN,p,γup in RN \Σhas positive solutions that blow up exactly at Σ.Remark 3.1.4. The constant AN,p,γ is chosen so that the model function uγ(x) =|x|− 2γp−1 is a singular solution to (3.3) that blows up exactly at the origin. In partic-ular,AN,p,γ = Λ(N−2γ2 − 2γp−1)for Λ(α) = 22γΓ(N+2γ+2α4 )Γ(N+2γ−2α4 )Γ(N−2γ−2α4 )Γ(N−2γ+2α4 ). (3.10)Note that, for the critical exponent p = N+2γN−2γ , the constant AN,p,γ coincides withΛN,γ =Λ(0), the sharp constant in the fractional Hardy inequality inRN . Its precisevalue is given in (3.44).Let us make some comments on the bibliography. First, the problem of unique-ness and non-degeneracy for some fractional ODE has been considered in [61, 97,98], for instance.The construction of singular solutions in the range of exponents for which theproblem is stable, i.e., NN−2γ < p < p1 for p1 <N+2γN−2γ defined in (3.12), was stud-ied in the previous paper by Ao, the author, Gonza´lez and Wei [8]. In addition,for the critical case p = N+2γN−2γ , solutions with a finite number of isolated singular-ities were obtained in the article by Ao, DelaTorre, Gonza´lez and Wei [9] usinga gluing method. The difficulty there was the presence of a non-trivial kernel forthe linearized operator. With all these results, together with Theorem 3.1.1, wehave successfully developed a complete fractional Mazzeo-Pacard program for theconstruction of singular solutions of the fractional Yamabe problem.Gluing methods for fractional problems are starting to be developed. A finitedimensional reduction has been applied in [62] to construct standing-wave solu-tions to a fractional nonlinear Schro¨dinger equation and in [82] to construct layeredsolutions for a fractional inhomogeneous Allen-Cahn equation.The next development came in [9] for the fractional Yamabe problem withisolated singularities, that we have just mentioned. There the model for an isolated97singularity is a Delaunay-type metric (see also [134, 135, 164] for the constructionof constant mean curvature surfaces with Delaunay ends and [133, 136] for thescalar curvature case). However, in order to have enough freedom parameters at theperturbation step, for the non-local gluing in [9] the authors replace the Delaunay-type solution by a bubble tower (an infinite, but countable, sum of bubbles). As aconsequence, the reduction method becomes infinite dimensional. Nevertheless, itcan still be treated with the tools available in the finite dimensional case and onereduces the PDE to an infinite dimensional Toda type system. The most recentworks related to gluing are [49, 51] for the construction of counterexamples to thefractional De Giorgi conjecture. This reduction is fully infinite dimensional.For the fractional De Giorgi conjecture with γ ∈ [12 ,1) we refer to [30, 38, 155]and the most recent striking paper [94]. Related to this conjecture, in the caseγ ∈ (0, 12) there exists a notion of non-local mean curvature for hypersurfaces inRn,see [41] and the survey [172]. Much effort has been made regarding regularity [17,31, 44] and various qualitative properties [79, 80]. More recent work on stabilityof non-local minimal surfaces can be found in [56]. Delaunay surfaces for thiscurvature have been constructed in [32, 33]. After the appearance of [58], Cabre´has pointed out that this paper also constructs Delaunay surfaces with constantnonlocal mean curvature.3.2 The fast decaying solutionWe aim to construct a fast-decay singular solution to the fractional Lane–Emdenequation(−∆RN )γu = AN,p,γup in RN \{0}. (3.11)for γ ∈ (0,1) and p in the range (3.9).We consider the exponent p1 = p1(N,γ) ∈ ( NN−2γ , N+2γN−2γ ) defined below by(3.12) such that the singular solution uγ(x) = |x|−2γp−1 is stable if and only ifNN−2γ < p < p1. In the notation of Remark 3.1.4, p1 as defined as the root ofpAN,p,γ = Λ(0). (3.12)The main result in this section is:98Proposition 3.2.1. For any ε ∈ (0,∞) there exists a fast-decay entire singular solu-tion uε of (3.11) such thatuε(x)∼O(|x|− 2γp−1)as |x| → 0,ε|x|−(N−2γ) as |x| → ∞.The proof in the stable case NN−2γ < p < p1 <N+2γN−2γ is already contained in thepaper [8], so we will assume for the rest of the section that we are in the unstableregimeNN−2γ < p1 ≤ p <N+2γN−2γ .We first prove uniqueness of minimal solutions for the non-local problem(3.14) using Schaaf’s argument and a fractional Pohozˇaev identity obtained byRos-Oton and Serra (Proposition 3.2.2 below). Then we perform a blow-upargument on an unbounded bifurcation branch. An application of Kelvin’stransform yields an entire solution of the Lane–Emden equation with the desiredasymptotics.Set A = AN,p,γ . Note that the Kelvin transform w(x) = |x|−(N−2γ)u(x|x|2)of usatisfies(−∆)γw(x) = A|x|βwp(x), (3.13)where β =: p(N−2γ)− (N+2γ) ∈ (−2γ,0).Consider the following non-local Dirichlet problem in the unit ball B1 =B1(0)⊂ RN , (−∆)γw(x) = λ |x|βA(1+w(x))p in B1,w = 0 in RN \B1.(3.14)Since (−∆)γ |x|β+2γ = c0|x|β and (−∆)γ(1−|x|2)γ+= c1 for some positive constantsc0 and c1, we have that |x|β+2γ+(1−|x|2)γ+ is a positive super-solution for small λ .Thus there exists a minimal radial solution wλ (r). Moreover, it is bounded, radiallynon-increasing for fixed λ ∈ (0,λ ∗) and non-decreasing in λ . We will show thatwλ is the unique solution of (3.14) for all small λ .99Proposition 3.2.2. There exists a small λ0 > 0 depending only on N ≥ 2 and γ ∈(0,1) such that for any 0 < λ < λ0, wλ is the unique solution to (3.14) among theclassC˜ 2γ (RN) ={w ∈ C 2(RN) :∫RN|w(x)|(1+ |x|)N+2γ dx < ∞}.The idea of the proof follows from [83] and similar arguments can be found in[161], [117] and [118].3.2.1 Useful inequalitiesThe first ingredient is the Pohozˇaev identity for the fractional Laplacian. Such iden-tities for integro-differential operators have been recently studied in [149], [151]and [114].Theorem 3.2.3 (Proposition 1.12 in [149]). Let Ω be a bounded C 1,1 domain,f ∈ C 0,1loc (Ω×R), u be a bounded solution of(−∆)γu = f (x,u) in Ω,u = 0 in RN \Ω, (3.15)and δ (x) = dist(x,∂Ω). Thenu/δ γ |Ω∈ C α(Ω) for some α ∈ (0,1),and there holds∫Ω(F(x,u)+1Nx ·∇xF(x,u)− N−2γ2N u f (x,u))dx=Γ(1+ γ)22N∫∂Ω( uδ γ)2(x ·ν)dσwhere F(x, t) =∫ t0 f (x,τ)dτ and ν is the unit outward normal to ∂Ω at x.Using integration by parts (see, for instance, (1.5) in [149]), it is clear that∫Ωu f (x,u)dx =∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx,100which yields our fundamental inequality:Corollary 3.2.4. Under the assumptions of Theorem 3.2.3, we have for any star-shaped domain Ω and any σ ∈ R,∫Ω(F(x,u)+1Nx ·∇xF(x,u)−σu f (x,u))dx≥(N−2γ2N−σ)∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx(3.16)The second ingredient is the fractional Hardy–Sobolev inequality which, viaHo¨lder inequality, is an interpolation of fractional Hardy inequality and fractionalSobolev inequality:Theorem 3.2.5 (Lemma 2.1 in [107]). Assume that 0≤α < 2γ <min{2,N}. Thenthere exists a constant c such thatc∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≥ (∫RN|x|−α |u| 2(N−α)N−2γ)N−2γN−α. (3.17)3.2.2 Proof of Proposition 3.2.2We are now in a position to prove the uniqueness of solutions of (3.14) with smallparameter.Proof. Suppose w and wλ are solutions to (3.14). Then u = w−wλ is a positivesolution to the Dirichlet problem(−∆)γu = λA|x|βgλ (x,u) in B1(0),u = 0 in RN \B1(0),where gλ (x,u) = (1+wλ (x)+u)p− (1+wλ (x))p ≥ 0 for u≥ 0. DenotingGλ (x,u) =∫ u0gλ (x, t)dt,101we apply (3.16) with f (x,u) = λA|x|βgλ (x,u) over Ω= B1 to obtain(N−2γ2N−σ)∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≤ λA∫B1(|x|βGλ (x,u)+1Nx ·∇x(|x|βGλ (x,u))−σ |x|βugλ (x,u))dx= λA∫B1|x|β((1+βN)Gλ (x,u)+1Nx ·∇xGλ (x,u)−σugλ (x,u))dx.(3.18)Note thatGλ (x,u) = u2∫ 10∫ 10pt(1+wλ (x)+ τtu)p−1 dτdt (3.19)and∇xGλ (x,u) = u2∫ 10∫ 10p(p−1)t(1+wλ (x)+ τtu)p−2 dτdt ·∇wλ (x).Since wλ is radially decreasing, x ·∇wλ (x)≤ 0 and hence x ·∇xGλ (x,u)≤ 0. Then(3.18) becomes(N−2γ2N−σ)∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≤ λA∫B1|x|β((1+βN)Gλ (x,u)−σugλ (x,u))dx.(3.20)Now, since for any λ ∈[0, λ∗2]and any x ∈ B1,limt→∞Gλ (x, t)tgλ (x, t)= limt→∞1p+1((1+wλ (x)+ t)p+1− (1+wλ (x))p+1)− (1+wλ (x))ptt ((1+wλ (x)+ t)p− (1+wλ (x))p)=1p+1,we deduce that for any ε > 0 there exists an M = M(ε)> 0 such thatGλ (x, t)≤1+ εp+1ugλ (x, t)102whenever t ≥M. From this we estimate the tail of the right hand side of (3.20) as∫B1∩{u≥M}|x|β((1+βN)Gλ (x,u)−σugλ (x,u))dx≤∫B1∩{u≥M}|x|β((1+βN)1+ εp+1−σ)ugλ (x,u)dx.We wish to choose ε and σ such that(1+βN)1+ εp+1< σ <N−2γ2N,so that the above integral is non-positive. Indeed we observe that(N+βN)1p+1− N−2γ2N=2(p(N−2γ)−2γ)− (N−2γ)(p+1)2N(p+1)=(p−1)(N−2γ)−4γ2N(p+1)< 0as p−1 ∈(2γN−2γ ,4γN−2γ). Then there exists a small ε > 0 such that(1+βN)1+ εp+1<N−2γ2N,from which the existence of such σ follows. With this choice of ε and σ , (3.20)gives (N−2γ2N−σ)∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≤ λA∫B1∩{u<M}|x|β((1+βN)Gλ (x,u)−σugλ (x,u))dx≤ λA(1+βN)∫B1∩{u<M}|x|βGλ (x,u)dx.Recalling the expression (3.19) for Gλ (x,u), we have(12− σN)∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≤ λACM ∫B1∩{u<M}|x|βu2 dx,103whereCM =p2(1+w λ∗2(0)+M)p−1(3.21)by the monotonicity properties of wλ .On the other hand, since p > NN−2γ ,−β =−p(N−2γ)+(N+2γ) = 2γ− (N−2γ)(p− NN−2γ)< 2γ,and thus the fractional Hardy–Sobolev inequality (3.17) impliesc∫RN∣∣∣(−∆) γ2 u∣∣∣2 dx≥ (∫RN|x|βu2η dx) 1η=(∫B1|x|βu2η dx) 1η,whereη =N+βN−2γ =p(N−2γ)−2γN−2γ = 1+(p− NN−2γ)> 1.Hence, (∫B1|x|βu2γ dx) 1γ≤ 2NN−2γ cCMλA∫B1|x|βu2 dx.However, by Ho¨lder’s inequality, we have∫B1|x|βu2 dx =∫B1|x| βη u2 · |x|β(1− 1η ) dx≤(∫B1|x|βu2η dx) 1η(∫B1|x|β dx)1− 1η≤ (N+β )−N+2γN+β(∫B1|x|βu2η dx) 1η.Therefore, we have(∫B1|x|βu2η dx) 1η≤ 2NcACM(N−2γ)(N+β )N+2γN+βλ(∫B1|x|βu2η dx) 1η,which forces u≡ 0 for anyλ < λ0 =(2NcACM(N−2γ)(N+β )N+2γN+β)−1. (3.22)1043.2.3 Existence of a fast-decay singular solutionConsider the space of twice differentiable, positive and radially decreasing func-tions supported in the unit ball,E ={w ∈ C 2(Rn) : w(x) = w˜(|x|), w˜′ ≤ 0, w > 0 in B1 and w≡ 0 in RN \B1}.We begin with an a priori estimate followed by a generic existence result for thenon-local ODE (3.14), from which a bifurcation argument follows.Lemma 3.2.6 (Uniform bound). There exists a universal constant C0 =C0(N,γ, p,λ ∗) such that for any function w ∈ E solving (3.14) and for anyx ∈ B1/2(0)\{0},w(x)≤C0|x|−β+2γp−1 =C0|x|−p(N−2γ)−Np−1 .Proof. Using the Green’s function for the Dirichlet problem in the unit ball ([148]),we havew(x) =∫B1G(x,y)λA|y|β (1+w(y))p dy,whereG(x,y) =C(N,γ)1|x− y|N−2γ∫ r0(x,y)0rγ−1(r+1)N2drwithr0(x,y) =(1−|x|2)(1−|y|2)|x− y|2 .Here C(N,γ) is some normalizing constant. Lety ∈ B |x|4(3x4)⊂ B |x|2(x)∩B|x|(0)⊂ B1(0).From y ∈ B |x|2(x), we have|x− y| ≤ |x|2≤ 14and |y| ≤ 3|x|2≤ 34105and sor0(x,y)≥(1− 14)(1− 916)116≥ 214> 5.On the other hand, since y ∈ B|x|(0) and u is radially non-increasing, we have|y|β ≥ |x|β and u(y)≥ u(x).Therefore, we may concludeG(x,y)≥C(N,γ)(2|x|)N−2γ ∫ 50rγ−1(r+1)N2drandw(x)≥ A∫B |x|4( 3x4 )C(N,γ)2N−2γ|x|N−2γ(∫ 50rγ−1(r+1)N2dr)λ0|x|βw(x)p dy≥C(N,γ)A2N−2γ(∫ 50rγ−1(r+1)N2dr)λ0 · |x|β|x|N−2γ w(x)p · |B1|( |x|4)N≥C−(p−1)0 |x|β+2γw(x)p,whereC−(p−1)0 =C(N,γ)|B1|Aλ02N+2γ∫ 50rγ−1(r+1)N2dr.The inequality clearly rearranges tow(x)≤C0|x|−β+2γp−1 ,as desired. The dependence of the constant C0 follows from (3.22) and (3.21).Lemma 3.2.7 (Existence). For any λ ∈ (0,+∞) the non-local Dirichlet problem(3.14) has a positive solution.Proof. We use Schauder fixed point theorem. Let us denote the Gagliardo norm by[u]2Hγ (RN) =∫RN∫RN|u(x)−u(y)|2|x− y|N+2γ dxdy.106Define the operator T byTw(x) =∫B1(0)G(x,y)λA|y|β (1+w(y))p dy for x ∈ B1(0)0 for x ∈ RN \B1(0),where G is the Green’s function as in the proof of Lemma 3.2.6.Suppose that w ∈ L2(B1(0)). We first observe that the right hand side of (3.14)is in L2NN+2γ (B1(0)), where 2NN+2γ is the conjugate of the critical Sobolev exponent2∗(N,γ) = 2NN−2γ . Indeed, by Lemma 3.2.6, we have|x|β (1+wp)≤ |x|β + |x|−2γw≤ |x|−N+ 2γp−1and the integrability follows from(−N+ 2γp−1)2NN+2γ+N =N(N−2γ)(p−1)(N+2γ)(4γN−2γ − (p−1))> 0.Using Ho¨lder inequality and fractional Sobolev inequality (see, for instance, [73]),we haveC1(N,γ)−1 ‖Tw‖2L2∗(N,γ)(B1(0)) ≤ [Tw]2Hγ (RN) =∫RNTw(x)(−∆)γTw(x)dx=∫B1(0)Tw(x) ·λA|x|β (1+w(x))p dx≤C2(N,γ, p,λ )‖Tw‖L2∗(N,γ)(B1(0))≤C3(N,γ, p,λ )[Tw]Hγ (RN).This implies the existence of C¯ > 0 such that‖Tw‖L2(B1(0)) ≤ C¯,i.e. T :B→B withB ={u ∈ L2(B1(0)) : ‖u‖L2(B1(0)) ≤ C¯},107as well as[Tw]Hγ (B1(0)) ≤ [Tw]Hγ (RN) ≤ C¯,hence the compactness of T via the Sobolev embedding. By Schauder fixed pointtheorem, there exists a weak solution w ∈ L2(B1(0)). It remains to apply ellipticregularity.Lemma 3.2.8 (Bifurcation). There exists a sequence of solutions (λ j,w j) of (3.14)in (0,λ ∗]×E such thatlimj→∞λ j = λ∞ ∈ [λ0,λ ∗] and limt→∞∥∥w j∥∥L∞ = ∞,where λ0 is given in Proposition 3.2.2.Proof. Consider the continuationC = {(λ (t),w(t)) : t ≥ 0}of the branch of minimal solutions {(λ ,wλ ) : 0≤ λ < λ ∗}, where (λ (0),w(0)) =(0,0). By Proposition 3.2.2, we see that C ⊂ (λ0,λ ∗]×E. Moreover, since wλ > 0in B1, we also have w > 0 in B1 for any (λ ,w) ∈ C . If C were bounded, thenLemma 3.2.7 would give a contradiction around limt→∞(λ (t),w(t)). Therefore, Cis unbounded and the existence of the desired sequence of pairs (λ j,w j) follows.We are ready to establish the existence of a fast-decay singular solution.Proof of Proposition 3.2.1. Let (λ j,w j) be as in Lemma 3.2.8. By Lemma 3.2.6,w j(x)≤C0|x|−β+2γp−1 .Definem j =∥∥w j∥∥L∞(B1) = w j(0) and R j = m p−1β+2γj = m p−1p(N−2γ)−Njso that m j, R j→ ∞ as j→ ∞. Set alsoWj(x) = λ1p−1 m−1j wλ j( xR j).108Then 0≤Wj ≤ 1 and Wj is a bounded solution to(−∆)γWj = mp−1j R−β−2γj A|x|β(λ1p−1j m−1j +Wj)pin BR j(0),Wj = 0 in RN \BR j(0),that is, (−∆)γWj = A|x|β(λ1p−1j m−1j +Wj)pin BR j(0),Wj = 0 in RN \BR j(0).In BR j(0), Wj(x) has the upper boundWj(x)≤ λ1p−1 m−1j ·C0(xR j)− β+2γp−1≤C0(λ1p−10 +(λ∗)1p−1)|x|− β+2γp−1=C1|x|−β+2γp−1 =C1|x|2γp−1−(N−2γ).(3.23)Note that |x|β ∈ Lq(BR j(0)) for any N2γ < q < N−β . Hence, for such q, by the regu-larity result in [150], Wj ∈ C ηloc(RN) for η = min{γ,2γ− Nq}∈ (0,1). Therefore,by passing to a subsequence, Wj converges uniformly on compact sets of RN to aradially symmetric and non-increasing function w which satisfies(−∆)γw = A|x|βwp in RN ,w(0) = 1,w(x)≤C1|x|2γp−1−(n−2γ),in view of (3.23).Now the family of rescaled solutions wε(x) = εw(εp−1β+2γ x)solves(−∆)γwε = A|x|βwpε in RN ,wε(0) = ε,wε(x)≤C1|x|2γp−1−(N−2γ) in RN \{0} .109Finally, its Kelvin transform uε(x) = |x|−(N−2γ)wε(x|x|2)satisfies(−∆)γuε = Aupε in RN \{0} ,uε(x)≤C1|x|−2γp−1 in RN \{0} ,uε(x)∼ ε|x|−(N−2γ) as |x| → ∞,as desired.3.3 The conformal fractional Laplacian in the presenceof k-dimensional singularities3.3.1 A quick review on the conformal fractional LaplacianHere we review some basic facts on the conformal fractional Laplacian that will beneeded in the next sections (see [52, 109] for the precise definitions and details).If (X ,g+) is a (n + 1)-dimensional conformally compact Einstein mani-fold (which, in particular, includes the hyperbolic space), one can define aone-parameter family of operators Pγ of order 2γ on its conformal infinityMn = ∂∞Xn+1. Pγ is known as the conformal fractional Laplacian and it can beunderstood as a Dirichlet-to-Neumann operator on M. In the particular case thatX is the hyperbolic space Hn+1, whose conformal infinity is M = Rn with theEuclidean metric, Pγ coincides with the standard fractional Laplacian (−∆Rn)γ .Let us explain this definition in detail. It is known that, having fixed a metricg0 in the conformal infinity M, it is possible to write the metric g+ in the normalform g+ = ρ−2(dρ2 + gρ) in a tubular neighborhood M× (0,δ ]. Here gρ is aone-parameter family of metrics on M satisfying gρ |ρ=0 = g0 and ρ is a definingfunction in X for the boundary M (i.e., ρ is a non-degenerate function such thatρ > 0 in X and ρ = 0 on M).Fix γ ∈ (0,n/2) not an integer such that n/2+ γ does not belong to the set ofL2-eigenvalues of −∆g+ . Assume also that the first eigenvalue for −∆g+ satisfiesλ1(−∆g+) > n2/4− γ2. It is well known from scattering theory [113, 115] that,110given w ∈ C ∞(M), the eigenvalue problem−∆g+W −(n22 − γ2)W = 0 in X , (3.24)has a unique solution with the asymptotic expansionW =W1ρn2−γ +W2ρn2+γ , W1,W2 ∈ C ∞(X) (3.25)and Dirichlet condition on MW1|ρ=0 = w. (3.26)The conformal fractional Laplacian (or scattering operator, depending on the nor-malization constant) on (M,g0) is defined taking the Neumann dataPg0γ w = dγW2|ρ=0, where dγ = 22γ Γ(γ)Γ(−γ) , (3.27)and the fractional curvature as Qg0γ := Pg0γ (1).Pg0γ is a self-adjoint pseudodifferential operator of order 2γ on M with the sameprincipal symbol as (−∆M)γ . In the case that the order of the operator is an eveninteger we recover the conformally invariant GJMS operators on M. In addition,for any γ ∈ (0, n2), the operator is conformal. Indeed,Pgwγ f = w− n+2γn−2γ Pg0γ (w f ), ∀ f ∈ C ∞(M), (3.28)for a change of metricgw := w4n−2γ g0, w > 0.Moreover, (3.28) yields the Qγ curvature equationPg0γ (w) = wn+2γn−2γ Qgwγ .Explicit formulas for Pγ are not known in general. The formula for the cylinderwill be given in Section 3.3.2, and it is one of the main ingredients for the lineartheory arguments of Section 3.7.111The extension (3.24) takes a more familiar form under a conformal change ofmetric.Proposition 3.3.1 ([52]). Fix γ ∈ (0,1) and g¯ = ρ2g+. Let W be the solutionto the scattering problem (3.24)-(3.25) with Dirichlet data (3.26) set to w. ThenW = ρ−n/2+γW is the unique solution to the extension problem −div(ρ1−2γ∇W)+Eg¯(ρ)W = 0 in (X , g¯),W |ρ=0= w on M,(3.29)where the derivatives are taken with respect to the metric g¯, and the zero-th orderterm is given byEg¯(ρ) =−∆g¯(ρ1−2γ2 )ρ1−2γ2 +(γ2− 14)ρ−(1+2γ)+ n−14n Rg¯ρ1−2γ= ρ−n2−γ−1{−∆g+−(n24 − γ2)}(ρn2−γ).(3.30)Moreover, we recover the conformal fractional Laplacian asPg0γ w =−d˜γ limρ→0ρ1−2γ∂ρW,whered˜γ =− dγ2γ =−22γ−1Γ(γ)γΓ(−γ) . (3.31)We also recall the following result, which allows us to rewrite (3.29) as a puredivergence equation with no zeroth order term. The more general statement can befound in Lemma 3.3.7, and it will be useful in the calculation of the Hamiltonianfrom Section 3.4.2.Proposition 3.3.2 ([47, 52]). Fix γ ∈ (0,1). LetW 0 be the solution to (3.24)-(3.25)with Dirichlet data (3.26) given by w ≡ 1, and set ρ∗ = (W 0) 1n/2−γ . The functionρ∗ is a defining function of M in X such that, if we define the metric g¯∗ = (ρ∗)2g+,then Eg¯∗(ρ∗)≡ 0. Moreover, ρ∗ has the asymptotic expansion near the conformalinfinityρ∗(ρ) = ρ[1+Qg0γ(n/2− γ)dγ ρ2γ +O(ρ2)].112By construction, if W ∗ is the solution to −div((ρ∗)1−2γ∇W ∗)= 0 in (X , g¯∗),W ∗= w on (M,g0),with respect to the metric g¯∗, thenPg0γ w =−d˜γ limρ∗→0(ρ∗)1−2γ ∂ρ∗W ∗+wQg0γ .Remark 3.3.3. In the particular case that X = Rn+1+ = {(x, `) : x ∈ Rn, ` > 0} ishyperbolic space Hn+1 with the metric g+ = d`2+|dx|2`2and M = Rn, this is justthe construction for the fractional Laplacian (−∆Rn)γ as a Dirichlet-to-Neumannoperator for a degenerate elliptic extension problem from [43]. Indeed, let U bethe solution to ∂``U +1−2γ`∂`U +∆RnU= 0 in Rn+1+ ,U |`=0= u on Rn,(3.32)then(−∆Rn)γu =−d˜γ lim`→0`1−2γ∂`U. (3.33)From now on, (X ,g+) will be fixed to be hyperbolic space with its standard metric.Our point of view in this chapter is to rewrite this extension problem (3.32)-(3.33)using different coordinates for the hyperbolic metric in X , such as (3.39).3.3.2 An isolated singularityBefore we go to the general problem, let us look at positive solutions to(−∆RN )γu = ΛN,γuN+2γN−2γ in RN \{0} (3.34)that have an isolated singularity at the origin. It is known ([40]) that such solutionshave the asymptotic behavior near the origin like r−N−2γ2 , for r = |x|. Thus it isnatural to writeu = r−N−2γ2 w. (3.35)113Note that the power of the nonlinearity in the right hand side of (3.34) is chosenso that the equation has good conformal properties. Indeed, let r = e−t and θ ∈SN−1 and write the Euclidean metric in RN as|dx|2 = dr2+ r2gSN−1in polar coordinates. We use conformal geometry to rewrite equation (3.34). Forthis, consider the conformal changeg0 :=1r2|dx|2 = dt2+gSN−1 ,which is a complete metric defined on the cylinder M0 :=R×SN−1. The advantageof using g0 as a background metric instead of the Euclidean one on RN is thefollowing: since the two metrics are conformally related, any conformal changemay be rewritten asg˜ = u4N−2γ |dx|2 = w 4N−2γ g0,where we have used relation (3.35). Then, looking at the conformal transformationproperty (3.28) for the conformal fractional Laplacian Pγ , it is clear thatPg0γ (w) = rN+2γ2 P|dx|2γ (r−N−2γ2 w) = rN+2γ2 (−∆RN )γu, (3.36)and thus equation (3.34) is equivalent toPg0γ (w) = ΛN,γwN+2γN−2γ in R×SN−1.The operator Pg0γ on R× SN−1 is explicit. Indeed, in [66] the authors calculateits principal symbol using the spherical harmonic decomposition for SN−1. Withsome abuse of notation, let µm, m = 0,1,2, . . . be the eigenvalues of ∆SN−1 , re-peated according to multiplicity (this is, µ0 = 0, µ1, . . . ,µN = N−1,. . . ). Then anyfunction on R×SN−1 may be decomposed as ∑m wm(t)Em, where {Em(θ)} is thecorresponding basis of eigenfunctions. The operator Pg0γ diagonalizes under sucheigenspace decomposition, and moreover, it is possible to calculate the Fourier114symbol of each projection. Letwˆ(ξ ) =1√2pi∫Re−iξ ·tw(t)dt (3.37)be our normalization for the one-dimensional Fourier transform.Proposition 3.3.4 ([66]). Fix γ ∈ (0, N2 ) and let Pmγ be the projection of the operatorPg0γ over each eigenspace 〈Em〉. ThenP̂mγ (wm) =Θmγ (ξ ) ŵm,and this Fourier symbol is given byΘmγ (ξ ) = 22γ∣∣∣Γ(12 + γ2 + 12√(N2 −1)2+µm+ ξ2 i)∣∣∣2∣∣∣Γ(12 − γ2 + 12√(N2 −1)2+µm+ ξ2 i)∣∣∣2 . (3.38)Proof. Let us give some ideas in the proof because they will be needed in the nextsubsections. It is inspired in the calculation of the Fourier symbol for the con-formal fractional Laplacian on the sphere Sn (see the survey [109], for instance).The method is, using spherical harmonics, to reduce the scattering equation (3.24)to an ODE. For this, we go back to the scattering theory definition for the frac-tional Laplacian and use different coordinates for the hyperbolic metric g+. Moreprecisely,g+ = ρ−2{ρ2+(1+ ρ24)2dt2+(1− ρ24)2gSN−1}, g¯ = ρ2g+, (3.39)where ρ ∈ (0,2), t ∈ R. The conformal infinity {ρ = 0} is precisely the cylinder(R×SN−1,g0). Actually, for the particular calculation here it is better to use thenew variable σ =− log(ρ/2), and writeg+ = dσ2+(coshσ)2dt2+(sinhσ)2gSN−1 . (3.40)115Using this metric, the scattering equation (3.24) is∂σσW +R(σ)∂σW +(coshσ)−2∂ttW +(sinhσ)−2∆SN−1W +(N24 − γ2)W = 0,(3.41)where W =W (σ , t,θ), σ ∈ (0,∞), t ∈ R, θ ∈ SN−1, andR(σ) =∂σ(coshσ sinhN−1σ)coshσ sinhN−1σ.After projection over spherical harmonics, and Fourier transform in t, the solutionto equation (3.41) maybe written asŴm = ŵmϕ(τ),where we have used the change of variable τ = tanh(σ) and ϕ := ϕ(m) is a solutionto the boundary value problem(1− τ2)∂ττϕ+(N−1τ − τ)∂τϕ+[−µm 1τ2 +(n24 − γ2) 11−τ2 −ξ 2]ϕ = 0,has the expansion (3.25) with w≡ 1 near the conformal infinity {τ = 1} ,ϕ is regular at τ = 0.This is an ODE that can be explicitly solved in terms of hypergeometric functions,and indeed,ϕ(τ) = (1+ τ)N4 − γ2 (1− τ)N4 − γ2 τ1−N2 +√(N2 −1)2+µm· 2F1(a,b;a+b− c+1;1− τ2)+S(1+ τ)N4 +γ2 (1− τ)N4 + γ2 τ1−N2 +√(N2 −1)2+µm· 2F1(c−a,c−b;c−a−b+1;1− τ2),(3.42)whereS(ξ ) =Γ(−γ)Γ(γ)∣∣∣Γ(12 + γ2 + 12√(N2 −1)2+µm+ ξ2 i)∣∣∣2∣∣∣Γ(12 − γ2 + 12√(N2 −1)2+µm+ ξ2 i)∣∣∣2 ,116anda = −γ2 +12 +12√(N2 −1)2+µm+ i ξ2 ,b = −γ2 +12 +12√(N2 −1)2+µm− i ξ2 ,c = 1+√(N2 −1)2+µm.The Proposition follows by looking at the Neumann condition in the expansion(3.25).The interest of this proposition will become clear in Section 3.7, where we cal-culate the indicial roots for the linearized problem. It is also the crucial ingredientin the calculation of the Green’s function for the fractional Laplacian with Hardypotential in Section 3.6.We finally recall the fractional Hardy’s inequality in RN ([18, 99, 124, 179])∫RNu(−∆RN )γudx≥ ΛN,γ∫RNu2r2γdx, (3.43)where ΛN,γ is the Hardy constant given byΛN,γ = 22γΓ2(N+2γ4 )Γ2(N−2γ4 )=Θ0γ(0). (3.44)Under the conjugation (3.35), inequality (3.43) is written as∫R×SN−1wPg0γ wdtdθ ≥ ΛN,γ∫R×SN−1w2 dtdθ .3.3.3 The full symbolNow we consider the singular Yamabe problem (3.5) in Rn \Rk. This particularcase is important because it is the model for a general higher dimensional singular-ity (see [121]).As in the introduction, set N := n− k. We define the coordinates z = (x,y),x ∈ RN , y ∈ Rk in the product space Rn \Rk = (Rn−k \{0})×Rk. Sometimes we117will consider polar coordinates for x, which arer = |x|= dist(·,Rk) ∈ R+, θ ∈ SN−1.We write the Euclidean metric in Rn as|dz|2 = |dx|2+ |dy|2 = dr2+ r2gSN−1 + |dy|2.Our model manifold M is going to be given by the conformal changegk :=1r2|dz|2 = gSN−1 +dr2+ |dy|2r2= gSN−1 +gHk+1 , (3.45)which is a complete metric, singular along Rk. In particular, M := SN−1×Hk+1.As in the previous case, any conformal change may be rewritten asg˜ = u4n−2γ |dz|2 = w 4n−2γ gk,where we have used relationu = r−n−2γ2 w,so we may just use gk as our background metric. As a consequence, arguing asin the previous subsection, the conformal transformation property (3.28) for theconformal fractional Laplacian yields thatPgkγ (w) = rn+2γ2 P|dz|2γ (r−n−2γ2 w) = rn+2γ2 (−∆Rn)γu, (3.46)and thus the original Yamabe problem (3.5) is equivalent to the following:Pgkγ (w) = Λn,γwn+2γn−2γ on M.Moreover, the expression for Pgkγ in the metric gk (with respect to the standardextension to hyperbolic space X = Hn+1) is explicit, and this is the statement ofthe following theorem. For our purposes, it will be more convenient to write the118standard hyperbolic metric asg+ = ρ−2{dρ2+(1+ ρ24)2gHk+1 + (1− ρ24 )2gSN−1} , (3.47)for ρ ∈ (0,2), so its conformal infinity {ρ = 0} is precisely (M,gk).Consider the spherical harmonic decomposition for SN−1 as in Section 3.3.2.Then any function w on M may be decomposed as w = ∑m wmEm, where wm =wm(ζ ) for ζ ∈ Hk+1. We show that the operator Pgkγ diagonalizes under such ei-genspace decomposition, and moreover, it is possible to calculate the Fourier sym-bol for each projection. Let ·̂ denote the Fourier-Helgason transform on Hk+1, asdescribed in the Appendix (section 3.11).Theorem 3.3.5. Fix γ ∈ (0, n2) and let Pmγ be the projection of the operator Pgkγ overeach eigenspace 〈Em〉. ThenP̂mγ (wm) =Θmγ (ξ ) ŵm,and this Fourier symbol is given byΘmγ (λ ) = 22γ∣∣∣Γ(12 + γ2 + 12√(N2 −1)2+µm+ λ2 i)∣∣∣2∣∣∣Γ(12 − γ2 + 12√(N2 −1)2+µm+ λ2 i)∣∣∣2 . (3.48)Proof. We follow the arguments in Proposition 3.3.4, however, the additional in-gredient here is to use Fourier-Helgason transform to handle the extra term ∆Hk+1that will appear.For the calculations below it is better to use the new variableσ =− log(ρ/2), ρ ∈ (0,2),and to rewrite the hyperbolic metric in Hn+1 from (3.47) asg+ = dσ2+(coshσ)2gHk+1 +(sinhσ)2gSN−1 ,119for the variables σ ∈ (0,∞), ζ ∈ Hk+1 and θ ∈ SN−1. The conformal infinity isnow {σ =+∞} and the scattering equation (3.24) is written as∂σσW +R(σ)∂σW +(coshσ)−2∆Hk+1W +(sinhσ)−2∆SN−1W +(n24 −γ2)W = 0,(3.49)where W =W (σ ,ζ ,θ), andR(σ) =∂σ((coshσ)k+1(sinhσ)N−1)(coshσ)k+1(sinhσ)N−1.The change of variableτ = tanh(σ), (3.50)transforms equation (3.49) into(1− τ2)2∂ττW +(n−k−1τ +(k−1)τ)(1− τ2)∂τW +(1− τ2)∆Hk+1W+( 1τ2 −1)∆SN−1W +(n24 − γ2)W = 0.Now we project onto spherical harmonics. This is, let Wm(τ,ζ ) be the projectionof W over the eigenspace 〈Em〉. Then each Wm satisfies(1−τ2)∂ττWm+(n−k−1τ +(k−1)τ)∂τWm+∆Hk+1Wm−µm 1τ2Wm+n24 −γ21−τ2 Wm = 0.(3.51)Taking the Fourier-Helgason transform in Hk+1 we obtain(1− τ2)∂ττŴm+(n−k−1τ +(k−1)τ)∂τŴm+[−µm 1τ2 +(n24 − γ2) 11−τ2 − (λ 2+ k24 )]Ŵm = 0for Ŵm = Ŵm(λ ,ω). Fixed m = 0,1, . . ., λ ∈ R and ω ∈ Sk, we know thatŴm = ŵmϕλk (τ),120where ϕ := ϕλk (τ) is the solution to the following boundary value problem:(1− τ2)∂ττϕ+(n−k−1τ +(k−1)τ)∂τϕ+[−µm 1τ2+(n24 − γ2) 11−τ2 − (λ 2+ k24 )]ϕ = 0,has the expansion (3.25) with w≡ 1 near the conformal infinity {τ = 1} ,ϕ is regular at τ = 0.(3.52)This is an ODE in τ that has only regular singular points, and can be explicitlysolved. Indeed, from the first equation in (3.52) we obtainϕ(τ) =A(1− τ2) n4− γ2 τ1− n2+ k2+√(n−k2 −1)2+µm 2F1(a,b;c;τ2)+B(1− τ2) n4− γ2 τ1− n2−√(n−k2 −1)2+µm 2F1(a˜, b˜; c˜; ,τ2),(3.53)for any real constants A,B, wherea = −γ2 +12 +12√(n−k2 −1)2+µm+ iλ2 ,a˜ = −γ2 +12 − 12√(n−k2 −1)2+µm+ iλ2 ,b = −γ2 +12 +12√(n−k2 −1)2+µm− iλ2 ,b˜ = −γ2 +12 − 12√(n−k2 −1)2+µm− iλ2 ,c = 1+√(n−k2 −1)2+µm,c˜ = 1−√(n−k2 −1)2+µm,and 2F1 denotes the standard hypergeometric function described in Lemma 3.10.1.Note that we can write λ instead of |λ | in the arguments of the hypergeometricfunctions because a = b, a˜ = b˜ and property (3.163).121The regularity at the origin τ = 0 implies B = 0 in (3.53). Moreover, using(3.162) we can writeϕ(τ) = A[α(1− τ2) n4− γ2 τ1− n2+ k2+√(n−k2 −1)2+µm 2F1(a,b;a+b− c+1;1− τ2)+β (1− τ2) n4+ γ2 τ1− n2+ k2+√(n−k2 −1)2+µm 2F1(c−a,c−b;c−a−b+1;1− τ2)],whereα =Γ(1+√(n−k2 −1)2+µm)Γ(γ)Γ(12+γ2+12√(n−k2 −1)2+µm−iλ2)Γ(12+γ2+12√(n−k2 −1)2+µm+iλ2) ,β =Γ(1+√(n−k2 −1)2+µm)Γ(−γ)Γ(12−γ2+12√(n−k2 −1)2+µm+iλ2)Γ(12−γ2+12√(n−k2 −1)2+µm−iλ2) .Note that our changes of variable giveτ = tanh(σ) =4−ρ24+ρ2= 1− 12ρ2+ · · · , (3.54)which yields, as ρ → 0,ϕ(ρ)∼ A[αρn2−γ +βρn2+γ + . . .].Here we have used (3.161) for the hypergeometric function.Looking at the expansion for the scattering solution (3.25) and the definition ofthe conformal fractional Laplacian (3.27), we must haveA = α−1, and Θmγ (λ ) = dγβα−1. (3.55)Property (3.165) yields (3.48) and completes the proof of Theorem 3.3.5.3.3.4 ConjugationWe now go back to the discussion in Section 3.3.2 for an isolated singularity butwe allow any subcritical power p ∈ ( NN−2γ , N+2γN−2γ ) in the right hand side of (3.34);122this is,(−∆RN )γu = AN,p,γup in RN \{0}. (3.56)This equation does not have good conformal properties. But, given u ∈ C ∞(RN \{0}), we can consideru = r−N−2γ2 w = r−2γp−1 v, r = e−t ,and define the conjugate operatorP˜g0γ (v) := r−N−2γ2 + 2γp−1 Pg0γ(rN−2γ2 − 2γp−1 v)= r2γp−1 p(−∆RN )γu. (3.57)Then problem (3.56) is equivalent toP˜g0γ (v) = AN,p,γvp in R×SN−1,for some v = v(t,θ) smooth, t ∈ R, θ ∈ SN−1.This P˜gkγ can then be seen from the perspective of scattering theory, and thus becharacterized as a Dirichlet-to-Neumann operator for a special extension problemin Proposition 3.3.9, as inspired by the paper of Chang and Gonza´lez [52]. Notethe Neumann condition (3.76), which differs from the one of the standard fractionalLaplacian.In the notation of Section 3.3.2, we set X = HN+1 with the metric given by(3.39). Its conformal infinity is M = R×SN−1 with the metric g0. We would liketo repeat the arguments of Section 3.3 for the conjugate operator P˜g0γ . But thisoperator does not have good conformal properties. In any case, we are able todefine a new eigenvalue problem that replaces (3.24)-(3.25).More precisely, let W be the unique solution to the scattering problem (3.24)-(3.25) with Dirichlet data (3.26) set to w. We define the function V by the followingrelationrQ0W = V , Q0 :=−N−2γ2 + 2γp−1 , (3.58)Substituting into (3.24), the new scattering problem is−∆g+V +(4+ρ24ρ)−2 [−2Q0 ∂tV −Q20V ]− (N42 − γ2)V = 0 in X , (3.59)123Moreover, if we setV = ρN2 −γV1+ρN2 +γV2, (3.60)the Dirichlet condition (3.26) will turn intoV1|ρ=0 = v, (3.61)and the Neumann one (3.27) intodγV2|ρ=0 = P˜g0γ (v). (3.62)The following proposition is the analogous to Proposition 3.3.4 for P˜g0γ :Proposition 3.3.6. Fix γ ∈ (0, n2) and let P˜mγ be the projection of the operator P˜g0γover each eigenspace 〈Em〉. Then˜̂Pmγ (vm) = Θ˜mγ (ξ ) v̂m,and this Fourier symbol is given byΘ˜mγ (ξ ) = 22γΓ(12+γ2+√(N2 −1)2+µm2+12(Q0+ξ i))·Γ(12+γ2+√(N2 −1)2+µm2− 12(Q0+ξ i))·Γ(12− γ2+√(N2 −1)2+µm2+12(Q0+ξ i))−1·Γ(12− γ2+√(N2 −1)2+µm2− 12(Q0+ξ i))−1.(3.63)Proof. We write the hyperbolic metric as (3.40) using the change of variable σ =− log(ρ/2). The scattering equation for W is (3.49) in the particular case k = 0,n = N, and thus, we follow the arguments in the proof of Theorem 3.3.5. Setr = e−t and project over spherical harmonics as in (3.51), which yields∂σσWm+R(σ)∂σWm+(coshσ)−2∂ttWm− (sinhσ)−2µmWm+(N24 − γ2)Wm = 0(3.64)124forR(σ) =∂σ (coshσ sinhN−1σ)coshσ sinhN−1σ.Recall the relation (3.58) and rewrite the extension equation (3.64) in terms of eachprojection Vm of V . This gives∂σσVm+R(σ)∂σVm+(coshσ)−2 {∂ttVm+2Q0∂tVm +Q20Vm}−(sinhσ)−2µmVm+(N24 − γ2)Vm = 0.(3.65)Now we use the change of variable (3.50), and take Fourier transform (3.37) withrespect to the variable t. Then(1− τ2)∂ττ V̂m+(N−1τ − τ)∂τ V̂m+[−µm 1τ2+(N24 − γ2) 11−τ2 − (ξ − iQ0)2]V̂m = 0. (3.66)The Fourier symbol (3.63) is obtained following the same steps as in the proof ofTheorem 3.3.5. Note that the only difference is the coefficient of V̂m in (3.66).We note here than an alternative way to calculate the symbol is by taking Four-ier transform in relation P˜g0γ (v) = e−Q0tPg0γ (w), as follows:˜̂Pmγ vm(t) = P̂mγ wm(ξ − iQ0) =Θmγ (ξ − iQ0)wˆm(ξ − iQ0) =Θmγ (ξ − iQ0)vˆm(ξ ).Thus Θ˜mγ (ξ ) =Θmγ (ξ − iQ0), as desired.Now we turn to Proposition 3.3.2, and we show that there exists a very specialdefining function adapted to V .Lemma 3.3.7. Let γ ∈ (0,1). There exists a new defining function ρ∗ such that, ifwe define the metric g¯∗ = (ρ∗)2g+, thenEg¯∗(ρ∗) = (ρ∗)−(1+2γ)( 4ρ4+ρ2)2Q20,where Eg¯∗(ρ∗) is defined in (3.30). The precise expression for ρ∗ isρ∗(ρ) =[α−1( 4ρ4+ρ2)N−2γ2 2F1(γp−1 ,N−2γ2 − γp−1 ; N2 ;(4−ρ24+ρ2)2)]2/(N−2γ), (3.67)125ρ ∈ (0,2), whereα =Γ(N2 )Γ(γ)Γ(γ+ γp−1)Γ(N2 − γp−1) .The function ρ∗ is strictly monotone with respect to ρ , and in particular, ρ∗ ∈(0,ρ∗0 ) forρ∗0 := ρ∗(2) = α−2N−2γ . (3.68)Moreover, it has the asymptotic expansion near the conformal infinityρ∗(ρ) = ρ[1+O(ρ2γ)+O(ρ2)]. (3.69)Proof. The proof follows Lemma 4.5 in [52]. The scattering equation (3.24) forW is modified to (3.59) when we substitute (3.58), but the additional terms do notaffect the overall result. Then we know that, given v≡ 1 on M, (3.59) has a uniquesolution V 0 with the asymptotic expansionV 0 = V 01 ρN2 −γ +V 02 ρN2 +γ , V 01 ,V02 ∈ C ∞(X)and Dirichlet condition on M = R×SN−1V 01 |ρ=0 = 1. (3.70)Actually, from the proof of Proposition 3.3.6 and the modifications of Proposition3.3.4 we do obtain an explicit formula for such V 0. Indeed, from (3.53) and (3.55)for k = 0, n = N, m = 0, replacing iλ by Q0, we arrive atV 0(τ) = ϕ(τ) = α−1(1− τ2)N4 − γ2 2F1(γp−1 ,N−2γ2 − γp−1 ; N2 ;τ2).Finally, substitute in the relation between τ and ρ from (3.54) and setρ∗(ρ) = (V 0)1N/2−γ (ρ). (3.71)Then, recalling (3.30), for this ρ∗ we haveEg¯∗(ρ∗) = (ρ∗)−N2 −γ−1{−∆g+−(N24 − γ2)}(V 0) = (ρ∗)−(1+2γ)( 4ρ4+ρ2)2Q20,126as desired. Here we have used the scattering equation for V 0 from (3.59) and thefact that V 0 does not depend on the variable t.To show monotonicity, denote η :=(4−ρ24+ρ2)2 for η ∈ (0,1). It is enough tocheck thatf (η) := (1−η)N−2γ4 2F1(γp−1 ,N−2γ2 − γp−1 ; N2 ;η)is monotone with respect to η . From properties (3.163) and (3.164) of the Hyper-geometric function and the possible values for p in (3.9) we can assert thatddηf (η)=ddη((1−η)−N−2γ4 + γp−1 (1−η)N−2γ2 − γp−1 2F1(N−2γ2 − γp−1 , γp−1 ; N2 ;η))=(N−2γ4 − γp−1)(1−η)−N−2γ4 + γp−1−1(1−η)N−2γ2 − γp−1· 2F1(N−2γ2 − γp−1 , γp−1 ; N2 ;η)− 2N(N−2γ2 − γp−1)(N2 − γp−1)(1−η)N−2γ2 − γp−1−12F1(N−2γ2 − γp−1 +1, γp−1 ; N2 +1;η)< 0.Remark 3.3.8. For the Neumann condition, note that, by construction,P˜g0γ (1) = dγV 02 |ρ=0, (3.72)while from (3.57) and the definition of AN,p,γ from (3.10),P˜g0γ (1) = r2γp−1 p(−∆RN )γ(r−2γp−1 ) = AN,p,γ .The last result in this section shows that the scattering problem for V (3.59) canbe transformed into a new extension problem as in Proposition 3.3.2, and whoseDirichlet-to-Neumann operator is precisely P˜g0γ . For this we will introduce the newmetric on RN \{0}g¯∗ = (ρ∗)2g+, (3.73)127where ρ∗ is the defining function defined in (3.67), and let us denoteV ∗ = (ρ∗)−(N/2−γ)V . (3.74)Proposition 3.3.9. Let v be a smooth function on M = R×SN−1. The extensionproblem−divg¯∗((ρ∗)1−2γ∇g¯∗V ∗)− (ρ∗)−(1+2γ)(4ρ4+ρ2)22Q0 ∂tV ∗= 0 in (X , g¯∗),V ∗|ρ=0= v on (M,g0),(3.75)has a unique solution V ∗. Moreover, for its Neumann data,P˜g0γ (v) =−d˜γ limρ∗→0(ρ∗)1−2γ∂ρ∗(V ∗)+AN,p,γv. (3.76)Proof. The original scattering equation (3.24)-(3.25) was rewritten in terms of V(recall (3.58)) as (3.59)-(3.60) with Dirichlet condition V1|ρ=0 = v. Let us rewritethis equation into the more familiar form of Proposition 3.3.2. We follow the argu-ments in [52]; the difference comes from some additional terms that appear whenchanging to V .First use the definition of the classical conformal Laplacian for g+ (that hasconstant scalar curvature Rg+ =−N(N+1)),Pg+1 =−∆g+− N2−14 ,and the conformal property of this operator (3.28) to assure thatPg+1 (V ) = (ρ∗)N+32 Pg¯1 ((ρ∗)−N−12 V ).Using (3.74) we can rewrite equation (3.59) in terms of V ∗ asPg¯1 ((ρ∗)1−2γ2 V ∗)+(ρ∗)−3−2γ2{(4+ρ24ρ)−2 (−2Q0 ∂tV ∗−Q20V ∗)+ (γ2− 14)V ∗}= 0128or, equivalently, using that for ρ := ρ1−2γ2 ,ρ∆g¯∗(ρV ) = divg¯∗(ρ2∇g¯∗V )+ρV∆g¯∗(ρ),we have−divg¯∗((ρ∗)1−2γ∇g¯∗V ∗)+Eg¯∗(ρ∗)V ∗+(ρ∗)−(1+2γ)(4+ρ22ρ)−2 (−2Q0 ∂tV ∗−Q20V ∗)= 0,with Eg¯∗(ρ∗) defined as in (3.30). Finally, note that the defining function ρ∗ waschosen as in Lemma 3.3.7. This yields (3.75).For the boundary conditions, let us recall the asymptotics (3.69). The Dirich-let condition follows directly from (3.26) and the asymptotics. For the Neumanncondition, we recall the definition of ρ∗ from (3.71), soV ∗ = (ρ∗)−N2 +γV =VV 0=V1+ρ2γV2V 01 +ρ2γV02,and thus−d˜γ limρ→0ρ1−2γ∂ρV ∗ = dγ(V2V01 −V1V 02)∣∣ρ=0 = P˜g0γ v−AN,p,γv,where we have used (3.61) and (3.62) for V , and (3.70) and (3.72) for V 0. Thiscompletes the proof of the Proposition.3.4 New ODE methods for non-local equationsIn this section we use the conformal properties developed in the previous sectionto study positive singular solutions to equation(−∆RN )γu = AN,p,γup in RN \{0}. (3.77)The first idea is, in the notation of Section 3.3.4, to set v = r2γp−1 u and rewrite thisequation asP˜g0γ (v) = AN,p,γvp, in R×SN−1, (3.78)129and to consider the projection over spherical harmonics in SN−1,P˜mγ (vm) = AN,p,γ(vm)p, for v = v(t),While in Proposition 3.3.6 we calculated the Fourier symbol for P˜mγ , now we willwrite it as an integro-differential operator for a well behaved convolution kernel.The advantage of this formulation is that immediately yields regularity for vm as in[65].Now we look at the m = 0 projection, which corresponds to finding radiallysymmetric singular solutions to (3.77). This is a non-local ODE for u= u(r). In thesecond part of the section we define a suitable Hamiltonian quantity in conformalcoordinates in the spirit a classical second order ODE.3.4.1 The kernelWe consider first the projection m= 0. Following the argument in [65], one can usepolar coordinates to rewrite P˜0γ as an integro-differential operator with a new con-volution kernel. Indeed, polar coordinates x= (r,θ) and x¯= (r¯, θ¯) in the definitionof the fractional Laplacian (3.2) give(−∆RN )γu(x) = kN,γP.V.∫ ∞0∫SN−1r−2γp−1 v(r)− r¯− 2γp−1 v(r¯)|r2+ r¯2+2rr¯〈θ , θ¯〉|N+2γ2r¯N−1 dr¯ dθ¯ .After the substitutions r¯ = rs and v(r) = (1− s− 2γp−1 )v(r)+ s− 2γp−1 v(r), and recallingthe definition for P˜0γ from (3.57) we haveP˜0γ (v) = kN,γP.V.∫ ∞0∫SN−1s−2γp−1+N−1(v(r)− v(rs))|1+ s2−2s〈θ , θ¯〉|N+2γ2dsdθ¯ +Cv(r),whereC = kN,γP.V.∫ ∞0∫SN−1(1− s− 2γp−1 )sN−1|1+ s2−2s〈θ , θ¯〉|N+2γ2dsdθ¯ .130Using the fact that v≡ 1 is a solution, one gets that C = AN,p,γ . Finally, the changeof variables r = e−t , r¯ = e−t ′ yieldsP˜0γ (v)(t) = P.V.∫R˜K0(t− t ′)[v(t)− v(t ′)]dt ′+AN,p,γv(t) (3.79)for the convolution kernel˜K0(t) =∫SN−1kN,γe−( 2γp−1−N)t|1+ e2t −2et〈θ , θ¯〉|N+2γ2dθ¯= ce−(2γp−1−N−2γ2 )t∫ pi0(sinφ1)N−2(cosh t− cosφ1)N+2γ2dφ1,where φ1 is the angle between θ and θ¯ in spherical coordinates, and c is a positiveconstant that only depends on N and γ . From here we have the explicit expression˜K0(t) = ce−( 2γ pp−1 )t 2F1(N+2γ2 ,1+ γ;N2 ;e−2t), (3.80)for a different constant c.As in [65], one can calculate its asymptotic behavior, and we refer to this paperfor details:Lemma 3.4.1. The kernel ˜K0(t) is decaying as t→±∞. More precisely,˜K0(t)∼|t|−1−2γ as |t| → 0,e−(N−2γp−1 )|t| as t→−∞,e−2pγp−1 |t| as t→+∞.The main result in this section is that one obtains a formula analogous to (3.79)for any projection P˜mγ . However, we have not been able to use the previous argu-ment and instead, we develop a new approach using conformal geometry and thespecial defining function ρ∗ from Proposition 3.3.2.Set Q0 =2γp−1 − N−2γ2 . In the notation of Proposition 3.3.6 we have:131Proposition 3.4.2. For the m-th projection of the operator P˜g0γ we have the expres-sionP˜mγ (vm)(t) =∫R˜Km(t− t ′)[vm(t)− vm(t ′)]dt ′+AN,p,γvm(t),for a convolution kernel ˜Km on R with the asymptotic behavior˜Km(t)∼|t|−1−2γ as |t| → 0,e−(1+γ+√(N−22 )2+µm+Q0)t as t→+∞,e(1+γ+√(N−22 )2+µm−Q0)t as t→−∞.Proof. We first consider the case that p = N+2γN−2γ so that Q0 = 0, and look at theoperator Pg0γ (w) from Proposition 3.3.4. Let ρ∗ be the new defining function fromProposition 3.3.2 and write a new extension problem for w in the correspondingmetric g¯∗. In this particular case, we can use (3.67) to writeρ∗(ρ) =[α−1( 4ρ4+ρ2)N−2γ22F1(N−2γ4 ,N−2γ4 ,N2 ,(4−ρ24+ρ2)2)] 2N−2γ, α =Γ(N2 )Γ(γ)Γ(N4 +γ2)2.The extension problem for g¯∗ is−divg¯∗((ρ∗)1−2γ∇g¯∗W ∗)= 0 in (X , g¯∗),W ∗|ρ=0= w on (M,g0);notice that it does not have a zero-th order term. Moreover, for the Neumann data,Pg0γ (w) =−d˜γ limρ∗→0(ρ∗)1−2γ∂ρ∗(W ∗)+ΛN,γw.From the proof of Proposition 3.3.2 we know that W ∗ = (ρ∗)−(N/2−γ)W , whereW is the solution to (3.41). Taking the projection over spherical harmonics, andarguing as in the proof of Proposition 3.3.4, we have that Ŵm(τ,ξ ) = ŵm(ξ )ϕ(τ),and ϕ = ϕ(m)ξ is given in (3.42). Let us undo all the changes of variable, but let uskeep the notation ϕ(ρ∗) = ϕ(m)ξ (τ).132Taking the inverse Fourier transform, we obtain a Poisson formulaW ∗m(ρ∗, t) =∫RPm(ρ∗, t− t ′)wm(t ′)dt ′,wherePm(ρ∗, t) =1√2pi∫R(ρ∗)−(N/2−γ)ϕ(ρ∗)eiξ t dξ .Note that, by construction,∫RPm(ρ∗, t)dt = 1 for all ρ∗. Now we calculatelimρ∗→0(ρ∗)1−2γ∂ρ∗(W ∗m) = limρ∗→0(ρ∗)1−2γW ∗m(ρ∗, t)−W ∗m(0, t)ρ∗= limρ∗→0(ρ∗)1−2γ∫RPm(ρ∗, t− t ′)ρ∗[wm(t ′)−wm(t)]dt ′.This implies thatPmγ (wm)(t) =∫RKm(t− t ′)[wm(t)−wm(t ′)]dt ′+ΛN,γwm(t), (3.81)where the convolution kernel is defined asKm(t) = d˜γ limρ∗→0(ρ∗)1−2γPm(ρ∗, t)ρ∗.If we calculate this limit, the precise expression for ϕ from (3.42) yields thatKm(t) =1√2pi∫R(Θmγ (ξ )−ΛN,γ)eiξ t dξ , Km(−t) =Km(t).which, of course, agrees with Proposition 3.3.4.The asymptotic behavior for the kernel follows from the arguments in Section3.6, for instance. In particular, the limit as t → 0 is an easy calculation sinceStirling’s formula implies that Θmγ (ξ ) ∼ |ξ |2γ as ξ → ∞. For the limit as |t| → ∞we use that the first pole of the symbol happens at ±i(1+ γ +√(N−22 )2+µm) soit extends analytically to a strip that contains the real axis. We have:Km(t)∼|t|−1−2γ as |t| → 0,e−(1+γ+√(N−22 )2+µm)|t| as t→±∞.133Now we move on to P˜g0γ (v), whose symbol is calculated in Proposition 3.3.6.Recall that under the change w(t) = r−Q0v(t), we haveP˜g0γ (v) = e−Q0tPg0γ (eQ0tv).From (3.81), if we split eQ0tvm(t) = (eQ0t − eQ0t ′)vm(t)+ eQ0t ′vm(t), thenP˜g0γ (v)(t) =Cv(t)+∫R˜Km(t− t ′)(vm(t)− vm(t ′))dt ′for the kernel˜Km(t) =Km(t)e−Q0t =1√2pie−Q0t∫R(Θmγ (ξ )−ΛN,γ)eiξ t dξ ,and the constantC = ΛN,γ +∫RKm(t− t ′)(1− eQ0(t ′−t))dt ′.We have not attempted a direct calculation for the constant C. Instead, by notingthat v ≡ 1 is an exact solution to the equation P˜g0γ (v) = AN,p,γvp, we have thatC = AN,p,γ , and this completes the proof of the proposition.3.4.2 The Hamiltonian along trajectoriesNow we concentrate on positive radial solutions to (3.78). These satisfyP˜0γ (v) = AN,p,γvp, v = v(t). (3.82)We prove the existence of a Hamiltonian type quantity for (3.82), decreasing alongtrajectories when p is in the subcritical range, while this Hamiltonian remains con-stant in t for critical p. Monotonicity formulas for non-local equations in the formof a Hamiltonian have been known for some time ([36, 38, 98]). Our main innova-tion is that our formula (3.83) gives a precise analogue of the ODE local case (seeProposition 1 in [132], and the notes [163]), and hints what the phase portrait for vshould be in the non-local setting. We hope to return to this problem elsewhere.134Theorem 3.4.3. Fix γ ∈ (0,1) and p ∈ ( NN−2γ , N+2γN−2γ ). Let v = v(t) be a solutionto (3.82) and set V ∗ its extension from Proposition 3.3.9. Then, the HamiltonianquantityH∗γ (t) =AN,p,γd˜γ(−12v2+1p+1vp+1)+12∫ ρ∗00(ρ∗)1−2γ{−e∗1(∂ρ∗V ∗)2+ e∗2(∂tV ∗)2} dρ∗= : H1(t)+H2(t)(3.83)is decreasing with respect to t. In addition, if p = N+2γN−2γ , then H∗γ (t) is constantalong trajectories.Here we write, using Lemma 3.3.7, ρ as a function of ρ∗, ande∗ =(ρ∗ρ)2(1+ ρ24)(1− ρ24)N−1,e∗1 =(ρ∗ρ)−2e∗,e∗2 =(ρ∗ρ)−2(1+ ρ24)−2e∗.(3.84)The constants AN,p,γ and d˜γ are given in (3.10) and (3.31), respectively.Proof. In the notation of Proposition 3.3.9, let v be a function on M = R×SN−1only depending on the variable t ∈ R, and let V ∗ be the corresponding solution tothe extension problem (3.75). Then V ∗ =V ∗(ρ, t). Use thatdivg¯∗((ρ∗)1−2γ∇g¯∗V ∗) =1e∗∂ρ∗(e∗(ρ∗)−(1+2γ)ρ2∂ρ∗V ∗)+(ρ∗)1−2γ(ρ∗ρ)−2(1+ ρ24)−2∂ttV ∗,where e∗ = |√g¯∗| is given in (3.84), so equation (3.75) reads−∂ρ∗(e∗ρ2(ρ∗)−(1+2γ)∂ρ∗V ∗)− (ρ∗)1−2γe∗(ρ∗ρ)−2(1+ ρ24)−2∂ttV ∗− (ρ∗)−(1+2γ)e∗(4+ρ24ρ)−22(−N−2γ2 + 2γp−1)∂tV ∗ = 0.135We follow the same steps as in [66]: multiply this equation by ∂tV ∗ and integratewith respect to ρ∗ ∈ (0,ρ∗0 ), where ρ∗0 is given in (3.68). Using integration byparts in the first term, the regularity of the function V ∗ at ρ∗0 , and the fact that12∂t[(∂tV ∗)2]= ∂ttV ∗∂tV ∗ and 12∂t[(∂ρ∗V ∗)2]= ∂tρ∗(V ∗)∂ρ∗V ∗, it holdslimρ∗→0(∂t(V ∗)e∗(ρ∗)−(1+2γ)ρ2∂ρ∗V ∗)+∫ ρ∗00[12 e∗(ρ∗)−(1+2γ)ρ2∂t[(∂ρ∗V ∗)2]]dρ∗−∫ ρ∗00[12(ρ∗)1−2γe∗(ρρ∗)2(1+ ρ24)−2∂t[(∂tV ∗)2]]dρ∗−∫ ρ∗00[(ρ∗)−(1+2γ)e∗(4ρ4+ρ2)22(−N−2γ2 + 2γp−1)[∂tV ∗]2]dρ∗= 0.But, for the limit as ρ∗→ 0, we may use (3.76) and (3.82) to obtaind˜γ limρ∗→0((ρ∗)−(1+2γ)ρ2e∗∂tV ∗∂ρ∗V ∗)=[−P˜g0γ v+AN,p,γv]∂tv = AN,p,γ(v− vp)∂tv= AN,p,γ∂t(12 v2− 1p+1 vp+1).Then, for H(t) defined as in (3.83), we have∂t [H(t)]=−2∫ ρ∗00[(ρ∗)1−2γe∗(ρρ∗)−2(1+ ρ24)−2(−N−2γ2 + 2γp−1) [∂tV ∗]2] dρ∗≤ 0,which proves the result.1363.5 The approximate solution3.5.1 Function spacesIn this section we define the weighted Ho¨lder space C 2,αµ,ν (Rn \Σ) tailored for thisproblem, following the notations and definitions in Section 3 of [133]. Intuitively,these spaces consist of functions which are products of powers of the distance to Σwith functions whose Ho¨lder norms are invariant under homothetic transformationscentered at an arbitrary point on Σ.Despite the non-local setting, the local Fermi coordinates are still in use aroundeach component Σi of Σ. When Σi is a point, these are simply polar coordinatesaround it. In case Σi is a higher dimensional sub-manifold, let T iσ be the tubularneighbourhood of radius σ around Σi. It is well known that T iσ is a disk bundleover Σi; more precisely, it is diffeomorphic to the bundle of radius σ in the normalbundle N Σi. The Fermi coordinates will be constructed as coordinates in thenormal bundle transferred to T iσ via such diffeomorphism. Let r be the distanceto Σi, which is well defined and smooth away from Σi for small σ . Let also y bea local coordinate system on Σi and θ the angular variable on the sphere in eachnormal spaceNyΣi. We denote by BNσ the ball of radius σ inNyΣi. Finally we letx denote the rectangular coordinate in these normal spaces, so that r = |x|, θ = x|x| .Let u be a function in this tubular neighbourhood and define‖u‖T iσ0,α,0 = supz∈T iσ|u|+ supz,z˜∈T iσ(r+ r˜)α |u(z)−u(z˜)||r− r˜|α + |y− y˜|α +(r+ r˜)α |θ − θ˜ |α ,where z, z˜ are two points in T iσ and (r,θ ,y),(r˜, θ˜ , y˜) are their Fermi coordinates.We fix a R > 0 be large enough such that Σ⊂ B R2(0) in Rn. Hereafter the letterz is reserved to denote a point in Rn \Σ. For notational convenience let us also fixa positive function ρ ∈ C ∞(Rn \Σ) that is equal to the polar distance r in each T iσ ,and to |z| in Rn \BR(0).137Definition 3.5.1. The spaceC l,α0 (Rn\Σ) is defined to be the set of all u∈C l,α(Rn\Σ) for which the norm‖u‖l,α,0 = ‖u‖C l,α (Σcσ/2)+K∑i=1l∑j=0‖∇ ju‖C 0,α (T iσ )is finite. Here Σcσ/2 = Rn \⋃Ki=1T iσ/2.Let us define a weighted Ho¨lder space for functions having different behaviorsnear Σ and at ∞. With R > 0 fixed, for any µ,ν ∈ R we setC l,αµ (BR \Σ) = {u = ρµ u¯ : u¯ ∈ C l,α0 (BR \Σ)},C l,αν (Rn \BR) = {u = ρν u¯ : u¯ ∈ C l,α0 (Rn \BR)},and thus we can define:Definition 3.5.2. The space C l,αµ,ν(Rn \Σ) consists of all functions u for which thenorm‖u‖C l,αµ,ν= supBR\Σ‖ρ−µu‖l,α,0+ supRn\BR‖ρ−νu‖l,α,0is finite. The spaces C l,αµ,ν(RN \ {0}) and C l,αµ,ν(Rn \Rk) are defined similarly, interms of the (global) Fermi coordinates (r,θ) or (r,θ ,y) and the weights rµ , rν .Remark 3.5.3. From the definition of C l,αµ,ν , functions in this space are allowed toblow up like ρµ near Σi and decay like ρν at ∞. Moreover, near Σi, their derivativeswith respect to up to l-fold products of the vector fields r∂r,r∂y,∂θ blow up nofaster than ρµ while at ∞, their derivatives with respect to up to l-fold products ofthe vector fields |z|∂i decay at least like ρν .Remark 3.5.4. As it is customary in the analysis of fractional order operators, wewrite many times, with some abuse of notation, C 2γ+αµ,ν .3.5.2 Approximate solution with isolated singularitiesLet Σ= {q1, · · · ,qK} be a prescribed set of singular points. In the next paragraphswe construct an approximate solution to(−∆RN )γu = AN,p,γup in RN \Σ,138and check that it is indeed a good approximation in certain weighted spaces.Let u1 be the fast decaying solution to (3.77) that we constructed in Proposition3.2.1. Now consider the following rescalinguε(x) = ε−2γp−1 u1( xε)in RN \{0}. (3.85)Choose χd to be a smooth cut-off function such that χd = 1 if |x| ≤ d andχd(x) = 0 for |x| ≥ 2d, where d > 0 is a positive constant such that d < d0 =infi 6= j{dist(qi,q j)/2}. Let ε¯ = {ε1, · · · ,εK} be a K-tuple of dilation parameterssatisfying cε ≤ εi ≤ ε < 1 for i = 1, . . . ,K. Now define our approximate solutionbyu¯ε(x) =K∑i=1χd(x−qi)uεi(x−qi).Set alsofε := (−∆x)γ u¯ε −AN,p,γ u¯pε . (3.86)For the rest of the section, we consider the spaces C 0,αµ˜,ν˜ , where− 2γp−1 < µ˜ < 2γ and − (n−2γ)< ν˜ . (3.87)Lemma 3.5.5. There exists a constant C, depending on d, µ˜, ν˜ only, such that‖ fε‖C 0,αµ˜−2γ,ν˜−2γ ≤CεN− 2pγp−1 . (3.88)Proof. Using the definition of (−∆)γ in RN , one has(−∆x)γ(χiuεi)(x−qi)= kN,γP.V.∫RNχi(x−qi)uεi(x−qi)−χi(x˜−qi)uεi(x˜−qi)|x− x˜|N+2γ dx˜= χi(x−qi)(−∆x)γuεi(x−qi)+ kN,γP.V.∫RN(χi(x−qi)−χi(x˜−qi))uεi(x˜−qi)|x− x˜|N+2γ dx˜139for each i = 1, . . . ,K. Using the equation (3.11) satisfied by uεi we havefε(x) = AN,p,γK∑i=1(χi−χ pi )upεi(x−qi)+ kN,γK∑i=1P.V.∫RN(χi(x−qi)−χi(x˜−qi))uεi(x˜−qi)|x− x˜|N+2γ dx˜=: I1+ kN,γ I2.Let us look first at the term I1. It vanishes unless |x− qi| ∈ [d,2d] for some i =1, . . . ,K. But then, one knows from the asymptotic behaviour of uεi thatuεi(x) = O(ε− 2γp−1i∣∣∣x−qiεi∣∣∣−(N−2γ))= O(εN−2γ− 2γp−1 )|x−qi|−(N−2γ),so one hasI1(x)≤CεN−2γ−2γp−1 if |x−qi| ∈ [d,2d].For the second term I2 = I2(x), we fix i = 1, . . . ,K, and divide it into threecases: x ∈ Bd/2(qi), x ∈ B2d(qi) \Bd/2(qi) and x ∈ RN \B2d(qi). In the first case,x ∈ Bd/2(qi), without loss of generality, assume that qi = 0, soI2(x) = P.V.∫RN(χi(x)−χi(x˜))uεi(x˜)|x− x˜|N+2γ dx˜= P.V[∫Bd(0)· · ·+∫B2d\Bd(0)· · ·+∫RN\B2d(0)· · ·].∫{d<|x˜|<2d}uεi(x˜)|x− x˜|N+2γ−2 dx˜+∫{|x˜|>2d}uεi(x˜)|x− x˜|N+2γ dx˜.Hereafter “· · ·” carries its obvious meaning, replacing the previously written integ-rand. Using that |x− x˜| ≥ 12 |x˜| for |x˜|> 2d when |x|< d2 we easily estimateI2(x)≤ O(εN−2pγp−1 )+∫ ∞2duεi(r)r1+2γdr = O(εN−2pγp−1 ).140Next, if x ∈ B2d(qi)\Bd/2(qi),I2(x) = P.V.[∫Bd/4(qi)· · ·+∫B2d(qi)\Bd/4(qi)· · ·+∫RN\B2d(qi)· · ·]= O(∫ d4ε0εN−2pγp−1 u1(x˜)dx˜)+O(∫B2d(qi)\Bd/4(qi)εN−2pγp−1|x− x˜|N+2γ−2 dx˜)+O(∫RN\B2d(qi)uεi(x˜)|x˜|N+2γ dx˜)= O(εN−2pγp−1 ).Finally, if x ∈ RN \B2d(qi),I2(x) = P.V.[∫Bd(qi)· · ·+∫B2d\Bd(qi)· · ·+∫RN\B2d(qi)· · ·]= O(εN−2pγp−1 |x|−(N+2γ)).Combining all the estimates above we get a C 0µ˜−2γ,ν˜−2γ bound for a pair of weightssatisfying (3.87). But passing to C 0,αµ˜−2γ,ν˜−2γ is analogous and thus we obtain (3.88).3.5.3 Approximate solution in general caseFirst note that our ODE argument for u1 also yields a fast decaying positive solutionto the general problem(−∆Rn)γu = AN,p,γup in Rn \Rk. (3.89)that is singular along Rk. Recall that we have set N = n− k.Indeed, define u˜1(x,y) := u1(x), where z = (x,y) ∈ Rn−k ×Rk, and use theLemma below. For this reason, many times we will use indistinctly the notationsu1(z) and u1(x). Moreover, after a straightforward rescaling, the constant AN,p,γmay be taken to be one.Lemma 3.5.6. If u is defined on RN and we set u˜(z) := u(x) in Rn in the notationabove, then(−∆Rn)γ u˜ = (−∆RN )γu.141Proof. We compute, first evaluating the y-integral,(−∆Rn)γ u˜(z) = kn,γP.V.∫Rnu˜(z)− u˜(z˜)|z− z˜|n+2γ dz˜= kn,γP.V.∫Rk∫RNu(x)−u(x˜)[|x− x˜|2+ |y− y˜|2] n+2γ2dx˜dy˜= kn,γP.V.∫RNu(x)−u(x˜)|x− x˜|N+2γ dx˜∫Rk1(1+ |y˜|2) n+2γ2dy˜= (−∆RN )γu(x).Here we have usedkn,γ∫Rk1(1+ |y˜|2) n+2γ2dy˜ = kN,γ . (3.90)(See Lemma A.1 and Corollary A.1 in [50]).Now we turn to the construction of an approximate solution for (3.3). Let Σ be ak-dimensional compact sub-manifold in Rn. We shall use local Fermi coordinatesaround Σ, as defined in Section 3.5.1. Let Tσ be the tubular neighbourhood ofradius σ around Σ. For a point z ∈ Tσ , denote it by z = (x,y) ∈N Σ×Σ whereN Σ is the normal bundle of Σ. Let B a ball in N Σ. We identify Tσ with B×Σ.In these coordinates, the Euclidean metric is written as (see, for instance, [137])|dz|2 =(|dx|2 O(r)O(r) gΣ+O(r)),where |dx|2 is the standard flat metric in B and gΣ the metric in Σ. The volumeform reduces todz = dx√detgΣ+O(r).In the ball B we use standard polar coordinates r > 0, θ ∈ SN−1. In addition,near each q ∈ Σ, we will consider normal coordinates for gΣ centered at q. Aneighborhood of Σ 3 q is then identified with a neighborhood of Rk 3 0 with themetricgΣ = |dy|2+O(|y|2),142which yields the volume formdz = dxdy(1+O(r)+O(|y|2)). (3.91)Note that Σ is compact, so we can cover it by a finite number of small balls B.As in the isolated singularity case, we define an approximate solution as fol-lows:u¯ε(x,y) = χd(x)uε(x)where χR is a cut-off function such that χd = 1 if |x| ≤ d and χd(x) = 0 for |x| ≥ 2d.In the following we always assume d < σ2 . Letfε := (−∆Rn)γ u¯ε − u¯pε .Lemma 3.5.7. Assume, in addition to (3.87), that − 2γp−1 < µ˜ < min{γ− 2γp−1 , 12 −2γp−1}. Then there exists a positive constant C depending only on d, µ˜, ν˜ but inde-pendent of ε such that for ε  1,‖ fε‖C 0,αµ˜−2γ,ν˜−2γ ≤Cεq, (3.92)where q = min{ (p−3)γp−1 − µ˜, 12 − γ+ (p−3)γp−1 − µ˜,N− 2pγp−1}> 0.Proof. Let us fix a point z = (x,y) ∈ Tσ , i.e. |x| < σ . By the definition of thefractional Laplacian,(−∆z)γ u¯ε(z) = kn,γP.V.∫Rnu¯ε(z)− u¯ε(z˜)|z− z˜|n+2γ dz˜= kn,γ[P.V.∫Tσ· · ·+∫T cσ· · ·]=: I1+ I2.Note that in this neighborhood we can write u¯ε(z) := u¯ε(x).For I2, since u¯ε(x˜) = 0 when z˜ = (x˜, y˜) ∈T cσ , one hasI2 = kn,γ∫T cσu¯ε(x)− u¯ε(x˜)|z− z˜|n+2γ dz˜ = u¯ε(x)kn,γ∫T cσ1|z− z˜|n+2γ dz˜≤Cu¯ε(x),143so I2 = O(1)u¯ε(x) (the precise constant depends on σ ). Next, for I1, use normalcoordinates y˜ in Σ centered at y in a neighborhood {|y− y˜|< σ1} for some σ1 smallbut fixed. The constants will also depend on this σ1. We haveI1 = kn,γP.V.∫Tσu¯ε(x)− u¯ε(x˜)|z− z˜|n+2γ dz˜= kn,γP.V.[∫{|y−y˜|≤|x|β }∩Tσ· · ·+∫{σ1>|y−y˜|>|x|β }∩Tσ· · ·+∫{|y−y˜|>σ1}∩Tσ· · ·]=: kn,γ [I11+ I12+ I13],where β ∈ (0,1) is to be determined later. The main term will be I11; let us calculatethe other two. First, for I12 we recall the expansion of the volume form (3.91),and approximate |z− z˜|2 = |x− x˜|2 + |y− y˜|2 and dz˜ = dx˜dy˜ modulo lower orderperturbations. ThenI12 =∫{σ1>|y−y˜|>|x|β }∩Tσu¯ε(x)− u¯ε(x˜)|z− z˜|n+2γ dz˜=∫{σ1>|y−y˜|>|x|β }∫{|x−x˜|≤|x|β }∩Tσ· · ·+∫{σ1>|y−y˜|>|x|β }∫{|x−x˜|>|x|β }∩Tσ· · ·.∫{|x−x˜|≤|x|β }(u¯ε(x)− u¯ε(x˜))(∫{σ1>|y−y˜|>|x|β }1|z− z˜|n+2γ dy˜)dx˜+∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ(∫{|yˆ|> |x|β|x−x˜|} 1(1+ |yˆ|2) n+2γ2dyˆ)dx˜.We estimate the above integrals in dy˜. For instance, for the first term, we have usedthat ∫{σ1>|y−y˜|>|x|β }1|z− z˜|n+2γ dy˜≤∫{σ1>|y−y˜|>|x|β }1|y− y˜|n+2γ dy˜.∫{|y|≥|x|β }1|y|n+2γ dy.∫ ∞|x|βrk−1rn+2γdr . |x|−β (N+2γ),which yields,I12 .∫{|x−x˜|≤|x|β }(u¯ε(x)− u¯ε(x˜))|x|−β (N+2γ) dx˜+∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜.144Now, since |x− x˜|> |x|β implies that |x˜|> c0|x|β and |x− x˜| ∼ |x˜| for some c0 > 0independent of |x| small,I12 . |x|−2βγ u¯ε(x)+ |x|−β (N+2γ)∫{|x−x˜|≤|x|β }u¯ε(x˜)dx˜+∫{|x˜|>c0|x|β }u¯ε(x˜)|x˜|N+2γ dx˜+∫{|x˜|≥c0|x|β }u¯ε(x)|x˜|N+2γ .We conclude, using the definition of u¯ε and the rescaling (3.85), thatI12 . |x|−2βγ u¯ε(x)+ εN−2γp−1 |x|−β (N+2γ)∫{|x˜|≤ |x|βε} u1(x˜)dx˜+ ε−2pγp−1∫{|x˜|> |x|βε} u1(x˜)|x˜|N+2γ dx˜+ |x|−2γβ u¯ε(x).For I13, one hasI13 ≤C∫Tσ|u¯ε(x)|+ |u¯ε(x˜)|dx˜≤C(u¯ε(x)+ εN−2pγp−1 (1+ |x|)−(N−2γ)).We look now into the main term I11, for which we need to be more precise,I11 = P.V.∫{|y−y˜|≤|x|β }∫{|x˜|<σ}u¯ε(x)− u¯ε(x˜)|z− z˜|n+2γ dz˜= P.V.[∫{|y−y˜|≤|x|β }∫{|x−x˜|≤|x|β }∩{|x˜|<σ}· · ·+∫{|y−y˜|≤|x|β }∫{|x−x˜|>|x|β }∩{|x˜|<σ}· · ·]=: I111+ I112.145Let us estimate these two integrals. First, since for |x| small, |x− x˜|< |x|β impliesthat |x˜|< σ , we havekn,γ I111 = kn,γ P.V.∫{|y−y˜|≤|x|β }∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)|z− z˜|n+2γ dz˜= kn,γ P.V.∫{|y−y˜|≤|x|β }∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)[|x− x˜|2+ |y− y˜|2] n+2γ2· (1+O(|x˜|)+O(|y− y˜|))dx˜dy˜= (1+O(|x|β ))kn,γ P.V.∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ·∫{|y|≤ |x|β|x−x˜|}1(1+ |y|2) n+2γ2dydx˜= (1+O(|x|β ))kn,γ P.V.∫{|x−x˜|≤|x|β} u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ·[∫Rk1(1+ |y|2) n+2γ2dy−∫{|y|> |x|β|x−x˜|} 1(1+ |y|2) n+2γ2dy]dx˜.Recall relation (3.90), thenkn,γ I111 = (1+O(|x|β ))kN,γP.V.∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜+O(1)∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ( |x|β|x− x˜|)−(N+2γ)dx˜= (1+O(|x|β ))kN,γP.V.∫RNu¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜+O(1)∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜+O(1)∫{|x−x˜|≤|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ( |x|β|x− x˜|)−(N+2γ)dx˜.146Using the definition of the fractional Laplacian in RN ,kn,γ I111 = (1+O(|x|β ))(−∆x)γ u¯ε(x)+O(1)∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜+O(1)|x|−β (N+2γ)∫{|x−x˜|≤|x|β }(u¯ε(x)− u¯ε(x˜))dx˜.Now we use a similar argument to that of I12, which yields∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ dx˜.∫{|x˜|>|x|β }u¯ε(x)|x˜|N+2γ dx˜+∫{|x˜|>|x|β }u¯ε(x˜)|x˜|N+2γ dx˜. |x|−2βγ u¯ε(x)+ ε−2pγp−1∫{ |x|βε <|x˜|< σε} u1(x˜)|x˜|N+2γ dx˜and also,|x|−β (N+2γ)∫{|x−x˜|≤|x|β }(u¯ε(x)− u¯ε(x˜))dx˜. |x|−β (N+2γ)[|x|βN u¯ε(x)+ εN−2γp−1∫{|x˜|≤ |x|βε }u1(x˜)dx˜]. |x|−2βγ u¯ε(x)+ εN−2γp−1 |x|−β (N+2γ)∫{|x˜|≤ |x|βε }u1(x˜)dx˜.In conclusion, one haskn,γ I111 = (1+O(|x|β ))(−∆x)γ u¯ε(x)+O(1)[|x|−2βγ u¯ε(x)+ ε−2pγp−1∫{ |x|βε ≤|x˜|< σε }u1(x˜)|x˜|N+2γ dx˜+ εN− 2γp−1 |x|−β (N+2γ)∫{|x˜|≤ |x|βε }u1(x˜)dx˜].147Next, for I112 we calculate similarlyI112 .∫{|y−y˜|≤|x|β }∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)[|x− x˜|2+ |y− y˜|2] n+2γ2dx˜dy˜=∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ∫{|y|≤ |x|β|x−x˜|}1(1+ |y|2) n+2γ2dydx˜=∫{|x−x˜|>|x|β }u¯ε(x)− u¯ε(x˜)|x− x˜|N+2γ( |x|β|x− x˜|)kdx˜= |x|−2βγ u¯ε(x)+ ε−k−2pγp−1 |x|βk∫{|x˜|> |x|βε }u1(x˜)|x˜|n+2γ dx˜.Combining the estimates for I111, I112 and I12, I13 we obtain(−∆z)γ u¯ε(x)= (1+O(|x|β ))(−∆x)γ u¯ε(x)+O(1)[|x|−2βγ u¯ε(x)+ ε−2pγp−1∫{ |x|βε <|x˜|< σε }u1(x˜)|x˜|N+2γ dx˜+ εN− 2γp−1 |x|−β (N+2γ)∫{|x˜|< |x|βε }u1(x˜)dx˜+ ε−k−2pγp−1 |x|βk∫{|x˜|> |x|βε }u1(x˜)|x˜|n+2γ dx˜+O(εN− 2pγp−1 (1+ |x|)−(N−2γ))]= (1+O(|x|β ))(−∆x)γ u¯ε(x)+O(1)|x|−2βγ u¯ε(x)+R1.In order to estimate R1 we use the asymptotic behavior of u1(x) at 0 and ∞. Bydirect computation one sees thatR1(x) =|x|−β 2pγp−1 , if |x|β < ε,εN−2pγp−1 |x|−βN , if |x|β > ε.The choice β = 12 yields thatR1 = O(|x|−γ)u¯ε(x), and thus(−∆z)γ u¯ε(z) = (1+O(|x|β ))(−∆x)γ u¯ε(x)+O(1)|x|−γ u¯ε(x).Finally, recall that u¯ε(x) = χd(x)uε(x), then by the estimates in the previous sub-section (3.88), one has| fε(z)|. |x| 12 |(−∆x)γ u¯ε(x)|+ |x|−γ u¯ε(x)+E , (3.93)148where the weighted norm of E can be bounded by εN−2pγp−1 .For z∈Rn \T σ2, the estimate is similar to the isolated singularity case, we omitthe details here. Then we may conclude‖ fε‖C 0,αµ˜−2γ,ν˜−2γ ≤Cεq,where q = min{ (p−3)γp−1 − µ˜, 12 − γ + (p−3)γp−1 − µ˜,N − 2pγp−1}, and it is positive if− 2γp−1 < µ˜ < min{γ− 2γp−1 , 12 − 2γp−1}.Remark 3.5.8. In general, in terms of the local Fermi coordinates (x,y) around afixed z0 = (0,0) ∈ Σ, for u ∈ C α+2γµ˜,ν˜ (Rn \Σ), one has the following estimate:(−∆)γu = (−∆Rn\Rk)γu(x,y)+ |x|τ‖u‖∗for |x|  1, |y|  1, and some τ > µ˜ − 2γ . Indeed, similar to the estimates inLemma 3.5.7, except the main term in I111, in the estimates, it suffices to controlthe terms u(x˜) by ‖u‖∗|x|µ˜ .3.6 Hardy type operators with fractional LaplacianHere we give a formula for the Green’s function for the Hardy type operator in RN ,Lφ := (−∆RN )γφ −κr2γφ , (3.94)where κ ∈R. In the notation of Section 3.3.2, after the conjugation (3.36) we maystudy the equivalent operatorL˜w := e−N+2γ2 tL (eN−2γ2 tw) = Pg0γ w−κw on R×SN−1for φ = eN−2γ2 w. Consider the projections over spherical harmonics: for m= 0,1 . . .,let wm be a solution toL˜mw := Pmγ wm−κwm = hm on R. (3.95)149Recall Proposition 3.3.4. Then, in Fourier variables, equation (3.95) simply be-comes(Θmγ (ξ )−κ)wˆm = hˆm.The behavior of this equation depends on the zeroes of the symbol Θmγ (ξ )−κ . Inany case, we can formally writewm(t) =∫R1Θmγ (ξ )−κhˆm(ξ )eiξ t dξ =∫Rhm(s)Gm(t− t ′)dt ′, (3.96)where the Green’s function for the problem is given byGm(t) =∫Reiξ t1Θmγ (ξ )−κdξ .Let us make this statement rigorous in the stable case (this is, below the Hardyconstant (3.44)):Theorem 3.6.1. Let 0≤ κ < ΛN,γ and fix m = 0,1, . . .. Assume that the right handside hm in (3.95) satisfieshm(t) =O(e−δ t) as t→+∞,O(eδ0t) as t→−∞, (3.97)for some real constants δ ,δ0. It holds:i. The function 1Θmγ (z)−κ is meromorphic in z∈C. Its poles are located at pointsof the form τ j± iσ j and −τ j± iσ j, for j = 0,1, . . .. In addition, τ0 = 0, andτ j = 0 for j large enough. For such j, σ j is an increasing sequence with noaccumulation points.ii. If δ > 0 and δ0 ≥ 0, then a particular solution of (3.95) can be written aswm(t) =∫Rhm(t ′)Gm(t− t ′)dt ′ (3.98)whereGm(t) = d0e−σ0|t|+∞∑j=1d je−σ j|t| cos(τ j|t|) (3.99)150for some constants d j, j = 0,1, . . .. Moreover, Gm is an even C ∞ functionwhen t 6= 0 andwm(t) = O(e−δ t) as t→+∞, wm(t) = O(eδ0t) as t→−∞.(3.100)iii. Now assume only that δ +δ0 ≥ 0. If σJ < δ < σJ+1 (and thus δ0 >−σJ+1),then a particular solution iswm(t) =∫Rhm(t ′)G˜m(t− t ′)dt ′whereG˜m(t) =∞∑j=J+1d je−σ j|t| cos(τ j|t|). (3.101)Moreover, G˜m is an even C ∞ function when t 6= 0 and the same conclusionas in (3.100) holds.Remark 3.6.2. All solutions of the homogeneous problem L˜mw= 0 are of the formw(t) =C−0 e−σ0t +C+0∞∑j=1C−j e−σ jt cosτ jt+∞∑j=1C+j e+σ jt cosτ jtfor some real constants C−j ,C+j , j = 0,1, . . .. Thus we can see that the only solutionto (3.95), in both the cases ii. and iii., with decay as in (3.100) is precisely wm.We also look at the case when κ leaves the stability regime. In order to simplifythe presentation, we only consider the projection m = 0 and the equationL˜0w = h. (3.102)In addition, we assume that only the first pole leaves the stability regime, whichhappens if ΛN,γ < κ < Λ′N,γ for some Λ′N,γ . Then, in addition to the poles above,we will have two real poles τ0 and −τ0. Some study regarding Λ′N,γ will be givenin the next section but we are not interested in its explicit formula.Proposition 3.6.3. Let ΛN,γ < κ < Λ′N,γ . Assume that h decays like O(e−δ t) ast→ ∞, and O(eδ0t) as t→−∞ for some real constants δ ,δ0. It holds:151i. The function 1Θ0γ (z)−κ is meromorphic in z ∈C. Its poles are located at pointsof the form τ j± iσ j and −τ j± iσ j, for j = 0,1, . . .. In addition, σ0 = 0, andτ j = 0 for j large enough. For such j, σ j is an increasing sequence with noaccumulation points.ii. If δ > 0, δ0 ≥ 0, then a solution of (3.102) can be written asw0(t) =∫Rh(t ′)G0(t− t ′)dt ′, (3.103)whereG0(t) = d0 sin(τ0t)χ(−∞,0)(t)+∞∑j=1d je−σ j|t| cos(τ jt)for some constants d j, j = 0,1, . . .. Moreover, G0 is an even C ∞ functionwhen t 6= 0 and we have the same decay as in (3.100).iii. The analogous statements to Theorem 3.6.1, iii., and Remark 3.6.2 hold.Further study of fractional non-linear equations with critical Hardy potentialhas been done in [3, 76], for instance.DefineAm = 12 +γ2 +12√(N2 −1)2+µm, Bm = 12 − γ2 + 12√(N2 −1)2+µm. (3.104)and observe that the symbolΘmγ (ξ ) = 22γ∣∣Γ(Am+ ξ2 i)∣∣2∣∣Γ(Bm+ ξ2 i)∣∣2 = 22γΓ(Am+ξ2 i)Γ(Am− ξ2 i)Γ(Bm+ξ2 i)Γ(Bm− ξ2 i)can be extended meromorphically to the complex plane, which will be denoted byΘm(z) := 22γΓ(Am+ z2 i)Γ(Am− z2 i)Γ(Bm+ z2 i)Γ(Bm− z2 i) ,for z ∈ C.Remark 3.6.4. It is interesting to observe thatΘm(z) =Θm(−z).152Moreover, thanks to Stirling formula (expression 6.1.37 in [4])Γ(z)∼ e−zzz− 12 (2pi) 12 , as |z| → ∞ in |arg z|< pi, (3.105)one may check that for ξ ∈ R,Θm(ξ )∼ |m+ξ i|2γ , as |ξ | → ∞, (3.106)and this limit is uniform in m. Here the symbol ∼ means that one can bound onequantity, above and below, by constant times the other. This also shows that, forfixed m, the behavior at infinity is the same as the one for the standard fractionalLaplacian (−∆)γ .The following proposition uses this idea to study the behavior as |t| → 0. Re-call that the Green’s function for the fractional Laplacian (−∆R)γ in one spacedimension is preciselyG(t) = |t|−(1−2γ).We will prove that Gm has a similar behavior.Proposition 3.6.5. Let γ ∈ (0,1/2). Thenlim|t|→0Gm(t)|t|−(1−2γ) = cfor some positive constant c.Proof. Indeed, recalling (3.106), we havelim|t|→0∫R1Θm(ξ )−λ eiξ t dξ|t|−(1−2γ) = lim|t|→0∫R1|t|2γ[Θm( ζt )−λ]eiζ dζ= limt→0[∫{|ζ |>tδ }. . .+∫{|ζ |≤tδ }. . .]=: limt→0[I1+ I2]for some 2γ < δ < 1.For I1, we use Stirling’s formula (3.105) to estimateI1 ∼∫{|ζ |>tδ }cos(ζ )|ζ |2γ dζ → c as t→ 0,153while for I2,|I2| ≤∫{|ζ |≤tδ }1|t|2γ[Θm(0)−λ] dζ → 0 as t→ 0,as desired.Lemma 3.6.6. Define the function Φ(x,ξ ) = 22γ∣∣Γ(Am+x+ ξ2 i)∣∣2∣∣Γ(Bm+x+ ξ2 i)∣∣2 . Then:i. Fixed x > Bm, Ψ(x,ξ ) is a (strictly) increasing function of ξ > 0.ii. Ψ(x,0) is a (strictly) increasing function of x > 0.Proof. As in [66], section 7, one calculates using (3.169),∂ξ (logΘm(ξ )) = Im{ψ(Bm+ x+ξ2 i))−ψ(Am+ x+ξ2 i)}= c Im∞∑l=0(1l+Am+ x+ξ2 i− 1l+Bm+ x+ξ2 i)> 0,as claimed. A similar argument yields the monotonicity in x.Now we give the proof of Theorem 3.6.1. Before we consider the general case,let us study first when κ = 0, for which Gm can be computed almost explicitly. Fixm = 0,1, . . .. The poles of the function 1Θm(z) happen at points z ∈ C such that± z2i+Bm =− j, for j ∈ N∪{0},i.e, at points {±iσ j} forσ j := 2(Bm+ j), j = 0,1, . . . . (3.107)Then the integral in (3.101) can be computed in terms of the usual residue formula.Define the region in the complex planeΩ= {z ∈ C : |z|< R, Imz > 0}. (3.108)154A standard contour integration along ∂Ω gives, as R→ ∞, thatGm(t) = 2pii∞∑j=0Res(eizt1Θm(z), iσ j)= 2pii∞∑j=0e−σ jtc j, (3.109)where c j = c j(m) is the residue of the function 1Θm(z) at the pole iσ j. This argumentis valid as long as the integral in the upper semicircle tends to zero as R→∞. Thishappens when t > 0 since |eizt |= e−t Imz. For t < 0, we need to modify the contourof integration to Ω= {z ∈ C : |z|< R, Imz < 0}, and we have that, for t < 0,Gm(t) = 2pii∞∑j=0c jeσ jt ,which of course gives that Gm is an even function in t. In any case Gm is exponen-tially decaying as |t| → ∞ with speed given by the first pole |σ0|= 2Bm.In addition, recalling the formula for the residues of the Gamma function from(3.168), we have thatc j =122γΓ(2Bm+ j)Γ(Am−Bm− j)Γ(Am+Bm+ j) limz→iσ jΓ(Bm+z2 i)(z− iσ j)=222γΓ(1− γ+√(n2 −1)2+ j)Γ(γ− j)Γ((γ+√(n2 −1)2+ j)−i(−1) jj!for j ≥ 1, which yields the (uniform) convergence of the series (3.99) by Stirling’sformula (3.105).Now take a general 0 < κ < Λn,γ . The function eizt 1Θm(z)−κ is meromorphic inthe complex plane C. Moreover, if z is a root of Θm(z) = κ , so are −z, z¯ and −z¯.Let us check then that there are no poles on the real line. Indeed, the firststatement in Lemma 3.6.6 implies that is enough to show thatΘm(0)−κ > 0.But again, from the second statement of the lemma, Θm(0) > Θ0(0), so we onlyneed to look at the case m = 0. Finally, just note that Θ0(0) = ΛN,γ > κ .155Next, we look for poles on the imaginary axis. For σ > 0, Θm(iσ) =Ψ(−σ ,0)and this function is (strictly) decreasing in σ . Moreover,Ψ(0,0)=Θm(0)=ΛN,γ >κ . Let σ0 ∈ (0,+∞] be the first point where Θm(iσ0) = κ . Then ±iσ0 are poles onthe imaginary axis. Moreover, the first statement of Lemma 3.6.6 shows that thereare no other poles in the strip {z : | Im(z)| ≤ σ0}.Denote the rest of the poles by z j := τ j+ iσ j, τ j− iσ j,−τ j+ iσ j and−τ j− iσ j,j= 1,2, . . .. Here we take σ j >σ0 > 0, τ j ≥ 0. A detailed study of the poles is givenin the Section 3.6.4. In particular, for large j, all poles lie there on the imaginaryaxis, and their asymptotic behavior is similar to that of (3.107).Now we can complete the proof of statement ii. of Theorem 3.6.1. Since wehave shown that there is a spectral gap σ0 from the real line, it is possible to modifythe contour of integration in (3.109) to prove a similar residue formula: for t > 0,Gm(t)= 2piiRes(eizt1Θm(z)−κ , iσ0)+2pii∞∑j=1[Res(eizt1Θm(z)−κ ,τ j + iσ j)+Res(eizt1Θm(z)−κ ,−τ j + iσ j)]= 2piic0e−σ0t +4pii∞∑j=1c je−σ jt cos(τ jt),and for t < 0 it is defined evenly. Here c j = c j(m) is the residue of the function1Θm(z)−κ at the point τ j + iσ j; it can be easily shown that c j is purely imaginary.Moreover, the asymptotic behavior for this residue is calculated in (3.121); indeed,c j ∼C j−2γ . The convergence of the series is guaranteed.Next, we turn to the proof the decay statement (3.100). The main idea is tocontrol the asymptotic behavior of a multipole expansion according to the locationof the poles. We start with a simple lemma:Lemma 3.6.7. If f1(t) = O(e−a|t|) as t → ∞, f2(t) = O(e−a+t) as t → +∞ andf2(t) = O(ea−t) as t→−∞ for some a,a+ > 0, a− >−a, thenf1 ∗ f2(t) = O(e−min{a,a+}t) as t→+∞.156Proof. Indeed, for t > 0,| f1∗ f2(t)|=∣∣∣∫Rf1(t− t ′) f2(t ′)dt ′∣∣∣.∫ 0−∞e−a(t−t′)ea−t′dt ′+∫ t0e−a(t−t′)e−a+t′ds+∫ +∞tea(t−t′)e−a+t′dt ′.The lemma follows by straightforward computations.Remark 3.6.8. It is interesting to observe that a− is not involved in the decay ast → +∞. Moreover, by reversing the role of t and −t, it is possible to obtain theanalogous statement for t→−∞ with the obvious modifications.Assume that δ0 ≥ 0 and that σJ < δ ≤ σJ+1 for some J ≥ 0. Let us use theprevious lemma to estimate, for t > 0,∣∣∣∫R[Gm(t− t ′)−J∑j=0d je−σ j|t−t′| cos(τ j(t− t ′))]h(t ′)dt ′∣∣∣≤∫RO(e−σJ+1|t−t′|)|h(t ′)|dt ′ = O(e−δ t).(3.110)Let us now look at the term e−σ j|t−t ′| cos(τ j(t−t ′)), j = 0, . . . ,J, inside the integral.Lemma (3.6.7) would yield an asymptotic behavior e−σ jt as t → +∞. We willprovide an additional argument to improve this behavior, by showing a furthercancelation. Indeed, calculateϕ j(t): =∫Re−σ j|t−t′| cos(τ j(t− t ′))h(t ′)dt ′=∫ t−∞e−σ j(t−t′) cos(τ j(t− t ′))h(t ′)dt ′+∫ +∞teσ j(t−t′) cos(τ j(t− t ′))h(t ′)dt ′.(3.111)The first integral in the right hand side above can be rewritten using that, by Fred-holm theory, the following compatibility condition must be satisfied:0 =∫Reσ jt′cos(τ j(t− t ′))h(t ′)dt ′ =∫ t−∞. . .+∫ +∞t. . . , (3.112)157and this is rigorous because our growth assumptions on h. Thus (3.111) is reducedtoϕ j(t) =∫ +∞t[−e−σ j(t−t ′)+ eσ j(t−t ′)]cos(τ j(t− t ′))h(t ′)dt ′.This is the standard variation of constants formula to produce a particular solutionfor the second order ODEϕ ′′j (t) = σ2j ϕ j(t)−2σ jh(t),and in particular shows that ϕ j(t) decays like h(t) as t→+∞, which is O(e−δ t).In addition, for the case 0 < δ ≤ σ0,∣∣∣∫RGm(t− t ′)h(t ′)dt ′∣∣∣≤ ∫RO(e−σ0|t−t′|)|h(t ′)|dt ′ = O(e−δ t).Finally, reversing t→+∞ and t→−∞ yields the proof of statement ii. in Theorem3.6.1.Now we give the proof of statement iii., which is similar to the above, but withweaker assumptions on δ0. Fix m= 0,1, . . ., and drop the subindex m for simplicity.Assume, as in the previous case, that σJ < δ < σJ+1. Here we only have thatδ0 ≥ −σJ+1. Then the integrals in (3.112) are not finite and the argument forj = 0, . . . ,J does not work. Instead, we change our Fourier transform to integrateon a different horizontal line R+ iϑ . This is, for w = w(t), setw˜(ζ ) =1√2pi∫R+iϑe−iζ t w(t)dt =1√2pi∫Re−i(ξ+iϑ)t w(t)dt = wˆ(ξ + iϑ),whose inverse Fourier transform isw(t) =1√2pi∫R+iϑeiζ t w˜(ζ )dζ ,Moreover, in the new variable ζ = ξ + iϑ we have thath˜(ζ ) = P˜(m)(w)(ζ ) = (Θm(ζ )−κ)w˜(ζ ).158Inverting this symbol we obtain a particular solutionw(t) =∫R+iϑ1Θm(ζ )−κ h˜(ζ )eiζ t dζ =∫R+iϑh(t ′)G˜ (t− t ′)dt ′,forG˜ (t) =∫R+iϑeiζ t1Θm(ζ )−κ dζ .Replacing the contour of integration from (3.108) to ∂Ωρ forΩρ = {z ∈ C : |z|< R, Imz > ρ}yields, as R→ ∞, thatG˜ (t) = 2pii∞∑j=J+1Res(eizt1Θm(z)−κ , iσ j)= 2pii∞∑j=J+1e−σ jtc j,where, as above, c j is the residue of the function 1Θm(z)−κ at the pole τ j + iσ j.Assume that δ + δ0 > 0, and take ϑ ∈ (−δ0,δ ). Then the growth hypothesison h from (3.97) imply that h˜(ζ ) is well defined. If δ0 +δ = 0, taking ϑ = δ , wecan still justify this argument by understanding the Fourier transform in terms ofdistributions. Moreover, we have the expansion as t→+∞,∣∣∣∫RG˜ (t− t ′)h(t ′)dt ′∣∣∣≤ ∫RO(e−σJ+1|t−t′|)|h(t ′)|dt ′ = O(e−δ t).That is, the problematic terms in (3.110) do not appear any longer, and we havefound a different particular solution wm.This completes the proof of Theorem 3.6.1.3.6.1 Beyond the stability regimeNow we look at the proof of Proposition 3.6.3. As we have mentioned, in orderto simplify the presentation, we only consider the projection m = 0. Let ΛN,γ <κ < Λ′N,γ be the region where we have exactly two real poles at τ0 and −τ0, forτ > 0. For this, just note that, for real ξ > 0, Lemma 3.6.6 shows that Θ0(ξ ) is159an increasing function in ξ , and it is even. Denote the rest of the poles as in theprevious subsection, for j = 1,2, . . ..We proceed as in the proof of Theorem 3.6.1 and writew0(t) =∫R1Θ0(ξ )−κ hˆ(ξ )eiξ t dξ =∫Rh(t ′)G0(t− t ′)dt ′. (3.113)In this case we can still invert the operator, but one needs to regularize the contourintegration in order to account for the real poles in order to give sense to the integralin (3.113). Indeed, for ε > 0 small, let us calculateG ε0 (t) =∫Reiξ t1Θ0(ξ − εi)−κ dξ .The poles are now τ0 + εi and τ0− εi. Define the region Ω = {z ∈ C : |z− (τ +εi)|< R,Rez> 0}. A standard contour integration along ∂Ω gives, as R→∞, thatfor t > 0,G ε0 (t) = 2piicε0ei(τ0+εi)t +4pii∞∑j=1e−σεj t cos(τεj t)cεj ,wherecε0 = Res( 1Θ0(z− εi)−κ ,τ0+ εi).Taking the limit ε → 0,G ε0 (t)→ G0(t) = 2piic0eiτ0t +4pii∞∑j=1c je−σ jt cos(τ jt),for t > 0, and extended evenly to the real line.Let us simplify this formula. Using Fredholm theory, to have a solution ofequation (3.102), h must satisfy the compatibility condition0 = e−iτ0t∫Rh(t ′)eiτ0t′dt ′ =∫Rh(t− t ′)e−iτ0t ′ dt ′=∫ +∞0h(t− t ′)e−iτ0t ′ dt ′+∫ 0−∞h(t− t ′)e−iτ0t ′ dt ′.160Substitute this expression into the formula below∫ +∞0h(t− t ′)e−iτ0t ′ dt ′+∫ 0−∞h(t− t ′)eiτ0t ′ dt ′=∫ 0−∞h(t− t ′)e−iτ0t ′ −∫ 0−∞h(t− t ′)eiτ0t ′ dt ′=∫ +∞th(t ′)sin(τ0(t− t ′))dt ′.Arguing as in the proof of Theorem 3.6.1 we obtain ii. The only difference withthe stable case is that the j = 0 term in the summation in formula (3.110) needs tobe replaced by ∫ +∞tsin(τ0(t− t ′))h(t ′)dt ′.A similar argument yields iii. too.3.6.2 A-priori estimates in weighted Sobolev spacesFor s > 0, we define the norm in R×SN−1 given by‖w‖2s =∞∑m=0∫R(1+ξ 2+m2)2s|wˆm(ξ )|2 dξ . (3.114)These are homogeneous norms in the variable r = e−t , and formulate the Sobolevcounterpart to the Ho¨lder norms in RN \ {0} from Section 3.5.1. That is, forw∗(r) := w(t) and s integer we have‖w‖20 =∞∑m=0∫ ∞0|w˜∗m|2r−1 dr,‖w‖21 =∞∑m=0∫ ∞0(|w˜∗m|2+ |∂rw˜∗m|2r2)r−1 dr,‖w‖22 =∞∑m=0∫ ∞0(|w˜∗m|2+ |∂rw˜∗m|2r2+ |∂rrw˜∗m|r4)r−1 dr.161One may also give the corresponding weighted norms, for a weight of the typer−ϑ = eϑ t . Indeed, one just needs to modify the norm (3.114) to‖w‖2s,ϑ =∞∑m=0∫R+iϑ(1+ξ 2+m2)2s|wˆm(ξ )|2 dξ .For instance, in the particular case s = 1, this is‖w‖21,ϑ =∞∑m=0∫ ∞0(|w∗m|2r−2ϑ + |∂rw∗m|2r2−2ϑ )r−1 dr.Proposition 3.6.9. Let s ≥ 2γ , and fix ϑ ∈ R such that the horizontal line R+ iϑdoes not cross any pole τ(m)j ± iσ (m)j , j = 0,1, . . ., m = 0,1, . . .. If w is a solution toL˜w = h in R×SN−1of the form (3.98), then‖w‖s,ϑ ≤C‖h‖s−2γ,ϑfor some constant C > 0.Proof. We project over spherical harmonics w=∑m wmEm, where wm is a solutionto L˜mwm = hm. Assume, without loss of generality, that ϑ = 0, otherwise replacethe Fourier transform ·ˆ by ·˜ on a different horizontal line. In particular, wˆm(ξ ) =(Θm(ξ )−κ)−1hˆm(ξ ), and we simply estimate‖w‖2s =∞∑m=0∫R(1+ |ξ |2+m2)2s|Θm(ξ )−κ|2 |hˆm(ξ )|2 dξ≤C∞∑m=0∫R(1+ |ξ |2+m2)2s−4γ |hˆm(ξ )|2 dξ=C‖h‖2s−2γ ,where we have used that(1+ |ξ |2+m2)2s|Θm(ξ )−κ|2 ≤C(1+ |ξ |2+m2)2s−4γ ,which follows from (3.106).1623.6.3 An application to a non-local ODEThe following result is not needed in the proof of the main theorem, but we havedecided to include it here because it showcases a classical ODE type behavior fora non-local equation, and it motivates the arguments in Section 3.7.Assume that we are in the unstable case, i.e., the setting of Proposition 3.6.3.Proposition 3.6.10. Let q> 0 and fix a potential onRwith the asymptotic behaviorV (t) =κ+O(e−qt) as t→+∞,O(1) as t→−∞,for r = e−t . Then the space of radial solutions to equation(−∆)γu− Vr2γu = 0 in RN (3.115)that have a bound of the form |u(r)| ≤Cr− n−2γ2 is two-dimensional.Proof. Let u be one of such solutions, and write w= urN−2γ2 , w=w(t). By assump-tion, w is bounded on R. Moreover, w satisfies the equation P(0)w−V w= 0, whichwill be written asL˜0w = h, for h := (V −κ)w.Then we have the bounds for hh(t) =O(e−qt) as t→+∞,O(1) as t→−∞,so we take δ = q > 0, δ0 = 0, and apply Proposition 3.6.3. Then w must be of theformw(t) = w0(t)+C10 sin(τ0t)+C20 cos(τ0t)+∞∑j=1e−σ jt[C1j sin(τ jt)+C2j cos(τ jt)]+∞∑j=1eσ jt[D1j sin(τ jt)+D2j cos(τ jt)]163for some real constants C10 ,C20 ,C1j ,C2j ,D1j ,D2j , j = 1,2, . . ., and w0 is given by(3.103). The same proposition yields that w0 is decaying as O(e−δ t) when t→+∞,so we must have D1j ,D2j = 0 for j = 1,2, . . .. Moreover, as t→−∞, v0 is bounded,which implies that only C10 and C20 survive. Note also that the behavior as t→+∞implies that this combination is nontrivial, so this yields a two-dimensional familyof bounded solutions.This argument also implies that any other solution must decay exponentiallyas O(e−δ t) for t → +∞ (this is, C10 = C20 = 0). Then we can iterate statement iii.with δ = lq, l = 2,3, . . . and δ0 = 0, to show that w decays faster than any O(e−δ t),δ > 0, as t → +∞, which gives that u(r) decays faster than any polynomial, thisis |u(r)| = o(|r|a) for every a ∈ N. Next, we use a unique continuation result forequation (3.115) to show that u ≡ 0. In the stable case, unique continuation wasproved in [84] using a monotonicity formula, while in the stable case it followsfrom [152], where Carleman estimates were the crucial ingredient.Finally we remark that if, in addition, the potential satisfies a monotonicitycondition, one can give a direct proof of unique continuation using Theorem 1from [98]. Note that, however, in [98] the potential is assumed to be smooth at theorigin. But one can check that the lack of regularity of the potential at the origincan be handled by the higher order of vanishing of u.3.6.4 Technical resultsHere we give a more precise calculation of the poles of the function 1Θm(z)−κ . Forthis, given κ ∈ R, we aim to solve the equationΓ(α+ iz)Γ(α− iz)Γ(β + iz)Γ(β − iz) −κ = 0 (3.116)with |α−β |< 1 and β < α .Lemma 3.6.11. Letz = iR+ζwith |z|> R0 and R0 sufficiently large. Then the solutions to (3.116) are containedin balls of radius Cκ sin((α−β )pi)N 2(α−β ) around the points z= (N +β )i, withN = [R] andC depending solely on α and β .164Proof. First we note, by using the identity Γ(s)Γ(1− s) = pi/sin(pis), thatΓ(α−R+ iζ )Γ(β −R+ iζ ) =Γ(1−β +R− iζ )Γ(1−α+R− iζ )sin(pi(β −R+ iζ ))sin(pi(α−R+ iζ ))=Γ(1−β +R− iζ )Γ(1−α+R− iζ )sin(pi(β −δ + iζ ))sin(pi(α−δ + iζ )) ,where we have denotedδ = R− [R] .Then, Stirling’s formula (3.105) yields|Γ(1+ z)| ∼ |z|Rez e−(Imz)arg(z)e−Rez√2pi |z| 12 ,which implies∣∣∣∣Γ(1−β +R− iζ )Γ(α+R− iζ )Γ(1−α+R− iζ )Γ(β +R− iζ )∣∣∣∣∼ (R2+ζ 2)α−β eζ(arctan ζ1−β+R+arctanζα+R−arctan ζ1−α+R−arctan ζβ+R)e−2(α−β ).Sincearctanζ1−β +R + arctanζα+R− arctan ζ1−α+R − arctanζβ +R= arctanζ1−β+R − ζ1−α+R1+ ζ1−β+Rζ1−α+R+ arctanζα+R − ζβ+R1+ ζα+Rζβ+R∼−2arctan (α−β )ζR2+ζ 2∼−2(α−β )ζR2+ζ 2,we can estimate, for R2+ζ 2 sufficiently large,∣∣∣∣Γ(1−β +R− iζ )Γ(α+R− iζ )Γ(1−α+R− iζ )Γ(β +R− iζ )∣∣∣∣∼ (R2+ζ 2)α−β e−2 (α−β )ζ2R2+ζ2 e−2(α−β ).Therefore, for R2+ζ 2 > R20 with R0 sufficiently large, we have the boundC−1(R2+ζ 2)α−β ≤∣∣∣∣Γ(1−β +R− iζ )Γ(α+R− iζ )Γ(1−α+R− iζ )Γ(β +R− iζ )∣∣∣∣≤C(R2+ζ 2)α−β ,165where C depends only on α and β . Hence,κC(R2+ζ 2)α−β≤∣∣∣∣ sin(pi(β −δ + iζ ))sin(pi(α−δ + iζ ))∣∣∣∣≤ κC−1(R2+ζ 2)α−β ,and by writingδ − iζ = β + z˜,we conclude that necessarily|z˜| ≤ Cκ sin((α−β )pi)R2(α−β ),which implies that solutions to (3.116) lie atz = iR+ζ = i [R]+ iβ + iz˜ = [R]+β +O(Cκ sin((α−β )pi)[R]2 (α−β )),and this proves the Lemma.Next we writez = i(β +N )+ z˜with N sufficiently large (according to the previous lemma) natural number, andequation (3.116) readsΓ(α−β −N + iz˜)Γ(α+β +N − iz˜)Γ(−N + iz˜)Γ(2β +N − iz˜) −κ = 0. (3.117)SinceΓ(−N + iz˜) = (−1)N −1Γ(−iz˜)Γ(1+ iz˜)Γ(N +1− iz˜)andΓ(α−β −N + iz˜) = (−1)N −1Γ(β −α− iz˜)Γ(1+α−β + iz˜)Γ(N +1−α+β − iz˜) ,166we can writeΓ(α−β −N + iz˜)Γ(α+β +N − iz˜)Γ(−N + iz˜)Γ(2β +N − iz˜)=Γ(N +1− iz˜)Γ(α+β +N − iz˜)Γ(N +1−α+β − iz˜)Γ(2β +N − iz˜)Γ(β −α− iz˜)Γ(1+α−β + iz˜)Γ(1+ iz˜)Γ(−iz˜) .Using thatΓ(β −α− iz˜)Γ(1+α−β + iz˜)Γ(1+ iz˜)Γ(−iz˜) =sin(−piiz˜)sin(pi(β −α− iz˜)) ,as well as Stirling’s formula (3.105) to estimateΓ(N +1− iz˜)Γ(α+β +N − iz˜)Γ(N +1−α+β − iz˜)Γ(2β +N − iz˜)∼ (N − iz˜)N −iz˜(α+β +N −1− iz˜)α+β+N −1−iz˜(N −α+β − iz˜)N −α+β−iz˜(2β −1+N − iz˜)2β−1+N −iz˜· e−2(α−β )√(N − iz˜)(α+β +N −1− iz˜)(N −α+β − iz˜)(2β −1+N − iz˜)∼N 2(α−β )e−2i(α−β )z˜e−2(α−β ),we arrive at the relationsin(−piiz˜)sin(pi(β −α− iz˜))e−2i(α−β )z˜ ∼ κN 2(α−β )e−2(α−β ),which impliesz˜∼ ipiκ sin(pi(β −α))N 2(α−β )e−2(α−β ).In fact, it is easy to see from (3.117) and the estimates above that a purely imaginarysolution z˜ does exist and a standard fixed point argument in each of the balls in theprevious lemma would show that it is unique.Finally, the half-ball of radius R0 around the origin in the upper half-plane isa compact set. Since the function at the left hand side of (3.116) is meromorphic,there cannot exist accumulation points of zeros and this necessarily implies that thenumber of zeros in that half-ball is finite.167We conclude then that the set of solutions to (3.116) consists of a finite numberof solutions in a half ball of radius R0 around the origin in the upper half-planetogether with an infinite sequence of roots at the imaginary axis located atzN = i(β +N )+O(κ sin(pi(β −α))N 2(α−β )e−2(α−β ))for N > R0, (3.118)as desired.Now we consider the asymptotics for the residues. We defineg(z) :=Γ(α+ iz)Γ(α− iz)Γ(β + iz)Γ(β − iz) −κ.We will estimate the residue of the function 1g(z) at the poles zN when N is suf-ficiently large. Given the fact that the poles are simple and the function 1/g(z) isanalytic outside its poles, we haveRes( 1g(z),zN)= limz→zN((z− zN ) 1g(z))=1g′(zN ).Henceg′(z) =ddz(Γ(α+ iz)Γ(α− iz)Γ(β + iz)Γ(β − iz))=−i Γ′(β + iz)Γ2(β + iz)(Γ(α+ iz)Γ(α− iz)Γ(β − iz))+1Γ(β + iz)(Γ(α+ iz)Γ(α− iz)Γ(β − iz))′=: S1+S2.Notice that g(zN ) = 0 implies thatΓ(β + izN ) =Γ(α+ izN )Γ(α− izN )κΓ(β − izN )168and therefore,S1 =−i Γ′(β + izN )Γ2(β + izN )(Γ(α+ izN )Γ(α− izN )Γ(β − izN ))=−iκ Γ′(β + izN )Γ(β + izN )=−iκψ(β + izN ),where ψ(z) is the digamma function. We recall the expansion (3.169),ψ(z) =−γ+∞∑l=0( 1l+1 − 1l+z).In this section, γ denotes the Euler constant. ThenS1 = iκ(γ+∞∑l=0(1l+β + izN− 1l+1))=pie−2(α−β )isin(pi(α−β ))N2(α−β )+O(1),(3.119)where we have used the asymptotics of l + β + izN when l =N from (3.118).Next, using again (3.118) we estimateS2 =1Γ(β + iz)(Γ(α+ iz)Γ(α− iz)Γ(β − iz))′∣∣∣∣z=zN= κi(Γ′(α+ iz)Γ(α+ iz)− Γ′(α− iz)Γ(α− iz) +Γ′(β − iz)Γ(β − iz))= κi(ψ(α−β −N +O(N −2(α−β )))−ψ(α+β +N +O(N −2(α−β )))+ψ(2β +N +O(N −2(α−β )))).By using the relationsψ(1− z)−ψ(z) = pi cot(piz)ψ(z)∼ ln(z− γ)+2γ , as |z| → ∞, Rez > 0,169we conclude, asN → ∞,ψ(α−β −N +O(N −2(α−β )))= ψ(1−α+β +N +O(N −2(α−β )))+pi cot(pi(1−α+β +N +O(N −2(α−β ))))= ln(N )+O(1),and henceS2 = iκ lnN +O(1). (3.120)Putting together (3.119) and (3.120) we findS1+S2 =pie−2(α−β )isin(pi(α−β ))N2(α−β )+ iκ lnN +O(1),and henceRes( 1g(z),zN)=ipie−2(α−β )sin(pi(α−β ))N2(α−β )−κ lnN +O(1)= isin(pi(α−β ))e2(α−β )piN −2(α−β )+O(lnNN 4(α−β )) (3.121)asN → ∞.3.7 Linear theory - injectivityLet u¯ε be the approximate solution from the Section 3.5.1. In this section weconsider the linearized operatorLεφ := (−∆Rn)γφ − pu¯p−1ε φ , in Rn \Σ, (3.122)where Σ is a sub-manifold of dimension k (or a disjoint union of smoothk-dimensional manifolds), andLεφ := (−∆RN )γφ − pAN,p,γ u¯p−1ε φ , in RN \{q1, . . . ,qK}. (3.123)170For this, we first need to study the model linearizationL1φ := (−∆RN )γφ − pAN,p,γup−11 φ = 0 in RN \{0}. (3.124)We will show that any solution (in suitable weighted spaces) to this equation mustvanish everywhere (from which injectivity in Rn \Rk follows easily), and then wewill prove injectivity for the operator Lε .Let us rewrite (3.124) using conformal properties and the conjugation (3.36).If we definew = rN−2γ2 φ , (3.125)then this equation is equivalent toPg0γ (w)−V w = 0, (3.126)for the radial potentialV =V (r) = r2γ pAN,p,γup−11 . (3.127)The asymptotic behavior of this potential is easily calculated using Proposition3.2.1 and, indeed, for r = e−t ,V (t) =pAN,p,γ +O(e−q1t) as t→+∞,O(etq0) as t→−∞, (3.128)for q0 = (N−2γ)(p−1)−2γ > 0.Let γ ∈ (0,1). By the well known extension theorem for the fractional Lapla-cian (3.32)-(3.33), equation (3.124) is equivalent to the boundary reaction problem∂``Φ+1−2γ`∂`Φ+∆RNΦ= 0 in RN+1+ ,−d˜γ lim`→0`1−2γ∂`Φ= pAN,p,γup−11 Φ on RN \{0},where d˜γ is defined in (3.31) and Φ|`=0 = φ .171Keeping the notations of Section 3.3.2 for the spherical harmonic decomposi-tion of SN−1, by µm we denote the m-th eigenvalue for −∆SN−1 , repeated accordingto multiplicity, and by Em(θ) the corresponding eigenfunction. Then we can writeΦ= ∑∞m=0Φm(r, `)Em(θ), where Φm satisfies the following:∂``Φm+1−2γ`∂`Φm+∆RNΦm−µmr2Φm= 0 in RN+1+ ,−d˜γ lim`→0`1−2γ∂`Φm= pAN,p,γup−11 Φm on RN \{0},(3.129)or equivalently, from (3.126),Pmγ (w)−V w = 0, (3.130)for w = wm = rN−2γ2 φm, φm =Φm(·,0).3.7.1 Indicial rootsLet us calculate the indicial roots for the model linearized operator defined in(3.124) as r → 0 and as r → ∞. Recalling (3.128), L1 behaves like the Hardyoperator (3.94) with κ = pAN,p,γ as r → 0 and κ = 0 as r → ∞. Moreover, wecan characterize very precisely the location of the poles in Theorem 3.6.1 andProposition (3.6.3).Here we find a crucial difference from the local case γ = 1, where the Fouriersymbol for the m-th projection Θm(ξ )− κ is quadratic in ξ , implying that thereare only two poles. In contrast, in the non-local case, we have just seen that thereexist infinitely many poles. Surprisingly, even though L1 is a non-local operator,its behavior is controlled by just four indicial roots, so we obtain results analogousto the local case.For the statement of the next result, recall the shift (3.125).Lemma 3.7.1. For the operatorL1 we have that, for each fixed mode m= 0,1, . . . ,i. At r = ∞, there exist two sequences of indicial roots{σ˜ (m)j ± iτ˜(m)j − N−2γ2 }∞j=0 and {−σ˜(m)j ± iτ˜(m)j − N−2γ2 }∞j=0.172Moreover,γ˜±m :=±σ˜ (m)0 − N−2γ2 =−N−2γ2 ±[1− γ+√(N−22 )2+µm], m = 0,1, . . . ,and γ˜+m is an increasing sequence (except for multiplicity repetitions).ii. At r = 0, there exist two sequences of indicial roots{σ (m)j ± iτ(m)j − N−2γ2 }∞j=0 and {−σ(m)j ± iτ(m)j − N−2γ2 }∞j=0.Moreover,a) For the mode m = 0, there exists p1 with NN−2γ < p1 <N+2γN−2γ (and itis given by (3.12)), such that for NN−2γ < p < p1 (the stable case), theindicial roots γ±0 :=±σ (0)0 − N−2γ2 are real with− 2γp−1 < γ−0 <−N−2γ2 < γ+0 ,while if p1 < p <N+2γN−2γ (the unstable case), then γ±0 are a pair of com-plex conjugates with real part −N−2γ2 and imaginary part ±τ(0)0 .b) In addition, for all j ≥ 1,σ (0)j >N−2γ2 .c) For the mode m = 1,γ−1 :=−σ (1)0 − N−2γ2 =− 2γp−1 −1.Proof. First we consider statement ii. and calculate the indicial roots at r = 0. Re-calling the shift (3.125), let L1 act on the function r−N−2γ2 +δ , and consider insteadthe operator in (3.126). Because of Proposition 3.3.4, for each m = 0,1, . . ., theindicial root γm :=−N−2γ2 +δ satisfies22γΓ(Am+ δ2)Γ(Am− δ2)Γ(Bm+ δ2)Γ(Bm− δ2) = pAN,p,γ , (3.131)173where Am,Bm are defined in (3.104).Note that if δ ∈ C is a solution, then −δ and ±δ are also solutions. Let uswrite δ2 = α+ iβ , and denoteΦm(α,β ) = 22γΓ(Am+ δ2)Γ(Am− δ2)Γ(Bm+ δ2)Γ(Bm− δ2) .From the expression, one can see that Φm(α,0) and Φm(0,β ) are real functions.We first claim that on the αβ -plane, provided that |α| ≤ Bm, any solution of(3.131) must satisfy α = 0 or β = 0, i.e., δ must be real or purely imaginary.Observing that the right hand side of (3.131) is real and so is Φm(0,β ) for β 6= 0,the claim follows from the strict monotonicity of the imaginary part with respectto α , namely∂∂αIm(Φm(α,β ))=− i2∂∂α[Φm(α,β )−Φm(α,−β )]=∞∑j=0Im[1j+Am+α−iβ +1j+Am−α−iβ +1j+Bm+α+iβ +1j+Bm−α+iβ]= β∞∑j=0[1( j+Am+α)2+β 2+ 1( j+Am−α)2+β 2 −1( j+Bm+α)2+β 2− 1( j+Bm−α)2+β 2],(3.132)the summands being strictly negative since Am > Bm. If β 6= 0 and |α| ≤ Bm, it iseasy to see that the above expression is not zero. This yields the proof of the claim.Moreover, Φm(α,0) and Φm(0,β ) are even functions in α,β , respectively. Us-ing the properties of the digamma function again, one can check that∂Φm(α,0)∂α< 0 for α > 0 (3.133)and∂Φm(0,β )∂β> 0 for β > 0. (3.134)174Let us consider now the case m = 0. Using the explicit expression for AN,p,γfrom (3.10), then δ must be a solution ofΓ(N4 +γ2 +δ2)Γ(N4 +γ2 − δ2)Γ(N4 − γ2 + δ2)Γ(N4 − γ2 − δ2) = pΓ(N2 − γp−1)Γ( γp−1 + γ)Γ( γp−1)Γ(N2 − γ− γp−1) =: λ (p). (3.135)From the arguments in [8] (see also the definition of p1 in (3.12)), there exists aunique p1 satisfying NN−2γ < p1 <N+2γN−2γ such thatΦ0(0,0) = λ (p1), andΦ0(0,0)>λ (p) when NN−2γ < p < p1, and Φ0(0,0)< λ (p) when p1 < p <N+2γN−2γ .Assume first that NN−2γ < p < p1 (the stable case). Then from (3.134), weknow that there are no indicial roots on the imaginary axis. Next we consider thereal axis. Since Φ0(B0,0) = 0, by (3.133), there exists an unique root α∗ ∈ (0,B0)such that Φ0(±α∗,0) = λ (p). We now show that α∗ ∈ (0, 2γp−1 − N−2γ2 ). Note thatΦ0( 2γp−1 − N−2γ2 ,0)= (1− p)Γ(N2 − γp−1)Γ( γp−1 + γ)Γ( γp−1)Γ(N2 − γ− γp−1) < 0.We conclude using the monotonicity of Φ0(α,0) in α .Now we consider the unstable case, i.e., for p > p1. First by (3.133), thereare no indicial roots on the real axis. Then by (3.132), in the region |α| ≤ B0,if a solution exists, then δ must stay in the imaginary axis. Since Φ0(0,β ) isincreasing in β and limβ→∞Φ0(0,β ) = +∞, we get an unique β ∗ > 0 such thatΦ0(0,±β ∗) = λ (p).In the notation of Section 3.6, we denote all the solutions to (3.135) to beσ (0)j ± iτ(0)j and −σ (0)j ± iτ(0)j , such that σ j is increasing sequence, then from theabove argument, one has the following properties: σ(0)0 ∈(0, 2γp−1 − N−2γ2), τ(0)0 = 0, forNN−2γ < p < p1,σ (0)0 = 0, τ(0)0 ∈ (0,∞), for p1 ≤ p < N+2γN−2γ ,andσ (0)j > 2B0 =N−2γ2 for j ≥ 1.175For the next mode m = 1, one can check by direct calculation that α1 = 2γp−1 +1− N−2γ2 is a solution to (3.131). By the monotonicity (3.133), there are no otherreal solutions in (0,α1). This also implies that Φ1(0,0) > λ (p), by (3.134), thereare no solutions in the imaginary axis.Moreover, using the fact thatΦm(α,0) is increasing in m, andΦm(±Bm,0) = 0,we get a sequence of real solutions αm ∈ (0,Bm) for m ≥ 1 that is increasing.Moreover, from (3.132), one also has that in the region |α| ≤ Bm, all the solutionsto (3.131) are real.Then, denoting the solutions to (3.131) by σ (m)j ± iτ(m)j and −σ (m)j ± iτ(m)j form≥ 1, we conclude that:σ (1)0 =2γp−1 +1− N−2γ2 , {σ(m)0 } is increasing, τ(m)0 = 0.We finally consider statement i. in the Lemma and look for the indicial rootsofL1 at r =+∞. In this case, δ will satisfy the following equation:22γ∣∣∣Γ(12 + γ2 + 12√(N2 −1)2+µm+ δ2)∣∣∣2∣∣∣Γ(12 − γ2 + 12√(N2 −1)2+µm+ δ2)∣∣∣2 = 0.For each fixed m = 0,1, . . ., the indicial roots occur when12 − γ2 + 12√(N2 −1)2+µm± δ2 = j, for j = 0,−1,−2, . . . ,−∞,or±δ = (1− γ)+√(N2 −1)2+µm+2 j, j = 0,1,2, . . . .Thus, the indicial roots forL1 at r =+∞ are given by−N−2γ2 ±{(1− γ)+√(N2 −1)2+µm}±2 j, j = 0,1, . . . .This finishes the proof of the Lemma.1763.7.2 Injectivity ofL1 in the weighted space C2γ+αµ,ν1The arguments in this section rely heavily on the results from Section 3.6. We fixµ > Re(γ+0 )≥−N−2γ2, ν1 ≤min{0,µ}. (3.136)Proposition 3.7.2. Under the hypothesis (3.136), the only solution φ ∈C 2γ+αµ,ν1 (RN \{0}) of the equationL1φ = 0 is the trivial solution φ ≡ 0.Proof. We would like to classify solutions to the following equation:(−∆RN )γφ = pAN,p,γup−11 φ in RN \{0},or equivalently, (3.129) or (3.130) for each m = 0,1, . . ..Step 1: the mode m = 0. Define the constant τ = pAN,p,γ and rewrite equation(3.126) asP0γ (w)− τw = (V − τ)w =: h (3.137)for some w= w(t), h= h(t). We use (3.128) and the definition of w to estimate theright hand side,h(t) =O(e−(q1+µ+N−2γ2 )t) as t→+∞,O(e−(ν1+N−2γ2 )t) as t→−∞.We use Theorem 3.6.1 and Proposition 3.6.3. There could be solutions to the homo-geneous problem of the form e(σ j±iτ j)t , e(−σ j±iτ j)t . But these are not allowed by thechoice of weights µ , ν1 from (3.136) since µ+ N−2γ2 > σ(0)0 and ν1+N−2γ2 < σ(0)1(for this, recall statements a) and b) in Lemma 3.7.1).Now we apply iii. of Theorem 3.6.1 (or Proposition 3.6.3) with δ = q1 +ν + N−2γ2 > Re(γ+0 +N−2γ2 ) = σ(0)0 and δ0 = −(ν1 + N−2γ2 ) > −σ(0)0 . Obviously,δ +δ0 > 0. Assume that σJ < δ < σJ+1. Then we can find a particular solution w0(depending on J) such thatw0(t) = (e−δ t), as t→+∞,177so w will have the same decay.Now, by the definition of h in (3.137), we can iterate this process with δ =lq1+ν+ N−2γ2 , l ≥ 2, and the same δ0, to obtain better decay when t→+∞. As aconsequence, we have that w decays faster than any e−δ t as t → +∞, which whentranslated to φ means that φ = o(ra) as r→ 0 for every a ∈ N. The strong uniquecontinuation result of [84] (stable case) and [152] (unstable case) for the operatorP0γ −V implies that φ must vanish everywhere.Step 2: the modes m = 1, . . . ,N. Differentiating the equation (3.11) we getL1∂u1∂xm= 0.Since u1 only depends on r, we have ∂u1∂xm = u′1(r)Em, where Em =xm|x| . Using thefact that −∆SN−1Em = µmEm, the extension for u′1(r) to RN+1+ solves (3.129) witheigenvalue N−1, and w1 := r N−2γ2 u′1 satisfies Pmγ w−V w = 0. Note that u′1 decayslike r−(N+1−2γ) as r→ ∞ and blows up like r− 2γp−1−1 as r→ 0.We know that also φm solves (3.130). Assume it decays like r−(N+1−2γ) asr→ ∞ and blows up like rγ+m as r→ 0. Then we can find a non-trivial combinationof u′1 and φm that decays faster than r−(N+1−2γ) at infinity. Since their singularitiesat 0 cannot cancel, this combination is non-trivial.Now we claim that no solution to (3.130) can decay faster than r−(N+1−2γ) at∞, which is a contradiction and yields that φm = 0 for m = 1, . . . ,N.To show this claim we argue as in Step 1, using the indicial roots at infinity(namely −(N + 1− 2γ) and 1) and interchanging the role of +∞ and −∞ in thedecay estimate. Using the facts that the solution decays like rσ for some σ <−(N−2γ+1), i.e. σ + N−2γ2 < −N−2γ2 − 1 = −σ(1)0 and Re(γ+m )+N−2γ2 < σ(1)1 ,one can show that the solution is identically zero.Step 3: the remaining modes m≥N+1. We use an integral estimate involvingthe first mode which has a sign, as in [59, 60]. We note that, in particular, φ1(r) =−u′1(r) > 0, which also implies that its extension Φ1 is positive. In general, the178γ-harmonic extension Φm of φm satisfiesdiv(`1−2γ∇Φm)= µm`1−2γr2Φm in RN+1+ ,−d˜γ lim`→0`1−2γ∂`Φm= pup−11 φm on RN+1+ .We multiply this equation by Φ1 and the one with m = 1 by Φm. Their differencegives the equality(µm−µ1)`1−2γr2ΦmΦ1 =Φ1 div(`1−2γ∇Φm)−Φm div(`1−2γ∇Φ1)= div(`1−2γ(Φ1∇Φm−Φm∇Φ1)).Let us integrate over the region where Φm > 0. The functions are regular enoughnear x = 0 by the restriction (3.136). The boundary ∂ {Φm > 0} is decomposedinto a disjoint union of ∂ 0 {Φm > 0} and ∂+ {Φm > 0}, on which `= 0 and ` > 0,respectively. Hence0≤ d˜γ(µm−µ1)∫{Φm>0}ΦmΦ1r2dxd`=∫∂ 0{Φm>0}(φ1 lim`→0`1−2γ∂Φm∂ν−φm lim`→0`1−2γ∂Φ1∂ν)dx+∫∂+{Φm>0}`1−2γ(Φ1∂Φm∂ν−Φm ∂Φ1∂ν)dxd`.The first integral on the right hand side vanishes due to the equations Φ1 and Φmsatisfy. Then we observe that on ∂+ {Φm > 0}, one has Φ1 > 0, ∂Φm∂ν ≤ 0 andΦm = 0. This forces (using µm > µ1)∫{Φm>0}ΦmΦ1r2dxd`= 0,which in turn implies Φm ≤ 0. Similarly Φm ≥ 0 and, therefore, Φm ≡ 0 for m ≥N+1. This completes the proof of the Proposition 3.7.2.1793.7.3 Injectivity of L1 on C2γ+αµ,ν1In the following, we set N = n− k and consider more general equation (3.89). SetL1 = (−∆Rn)γ − pAN,p,γup−11 in Rn \Rk.Proposition 3.7.3. Choose the weights µ,ν1 as in Proposition 3.7.2. The only solu-tion φ ∈ C 2γ+αµ,ν1 (Rn \Rk) of the linearized equation L1φ = 0 is the trivial solutionφ ≡ 0.Proof. The idea is to use the results from Section 3.3.3 to reduce L1 to the simplerL1, taking into account that u1 only depends on the variable r but not on z. In thenotation of Proposition 3.3.4, define w = r−N−2γ2 φ , and wm its m-th projection overspherical harmonics. Set wˆm(λ ,ω), λ ∈ R, ω ∈ Sk to denote its Fourier-Helgasontransform. By observing that the full symbol (3.48), for each fixed ω , coincideswith the symbol (3.38), we have reduced our problem to that of Proposition 3.7.2.This completes the proof.3.7.4 A priori estimatesNow we go back to the linearized operator Lε from (3.123) for the point singularitycase RN \{q1, . . . ,qK}, or (3.122) for the general Rn \Σ, and consider the equationLεφ = h. (3.138)For simplicity, we use the following notation for the weighted norms‖φ‖∗ = ‖φ‖C 2γ+αµ,ν , ‖h‖∗∗ = ‖h‖C 0,αµ−2γ,ν−2γ . (3.139)Moreover, for this subsection, we assume that µ,ν satisfyRe(γ+0 )< µ ≤ 0, −(n−2γ)< ν .For this choice of weights we have the following a priori estimate:180Lemma 3.7.4. Given h with ‖h‖∗∗ < ∞, suppose that φ be a solution of (3.138),then there exists a constant C independent of ε such that‖φ‖∗ ≤C‖h‖∗∗.Proof. We will argue by contradiction. Assume that there exists ε j → 0, and asequence of solutions {φ j} to Lε jφ j = h j such that‖φ j‖∗ = 1, and ‖h j‖∗∗→ 0 as j→ ∞.In the following we will drop the index j without confusion.We first consider the point singularity case RN \Σ for Σ = {q1, . . . ,qK}. ByGreen’s representation formula one hasφ(x) =∫RNG(x, x˜)[h+ pAN,p,γ u¯p−1ε φ ]dx˜ =: I1+ I2,where G is the Green’s function for the fractional Laplacian (−∆RN )γ given byG(x, x˜) =CN,γ |x− x˜|−(N−2γ) for some normalization constant CN,γ . In the first step,let x ∈ RN \⋃i Bσ (qi). In this caseI1 .∫RN|x− x˜|−(N−2γ)h(x˜)dx˜=∫{dist(x˜,Σ)< σ2 }· · ·+∫{ σ2 <dist(x˜,Σ)< |x|2 }· · ·+∫{ |x|2 <dist(x˜,Σ)<2|x|}· · ·+∫{dist(x˜,Σ)>2|x|}· · ·≤C‖h‖∗∗(|x|−(N−2γ)+ |x|ν)≤C‖h‖∗∗|x|ν ,because of our restriction of ν . Moreover,I2 =∫G(x, x˜)pu¯ε(x˜)p−1φ(x˜)dx˜=∫{dist(x˜,Σ)<ε}· · ·+∫{ σ2 >dist(x˜,Σ)>ε}· · ·+∫{dist(x˜,Σ)> σ2 }· · ·=: I21+ I22+ I23.181CalculateI21 ≤∫{dist(x˜,Σ)<ε}|x− x˜|−(N−2γ)ρ(x˜)−2γφ≤ ‖φ‖∗∫{dist(x˜,Σ)<ε}|x− x˜|−(N−2γ)ρ(x˜)µ−2γ dx˜,≤CεN+µ−2γ‖φ‖∗ρ(x)−(N−2γ),where ρ is the weight function defined in Section 3.5.1, andI22 =∫{dist(x˜,Σ)> σ2 }|x− x˜|−(N−2γ)εN(p−1)−2pγρ(x˜)−(N−2γ)(p−1)φ dx˜≤ ‖φ‖∗∫{R>dist(x˜,Σ)> σ2 }|x− x˜|−(N−2γ)εN(p−1)−2pγρ(x˜)µ−(N−2γ)(p−1) dx˜+‖φ‖∗∫{dist(x˜,Σ)>R}|x− x˜|−(N−2γ)εN(p−1)−2pγρ(x˜)ν−(N−2γ)(p−1)]dx˜. εN(p−1)−2pγρ(x)−(N−2γ)‖φ‖∗.Next for I23,I23 .∫{ε<dist(x˜,Σ)< σ2 }|x− x˜|−(N−2γ)εN(p−1)−2pγρ(x˜)−(N−2γ)(p−1)φ dx˜≤ εN(p−1)−2pγρ(x)−(N−2γ)‖φ‖∗∫{ε<|x˜|< σ2 }|x˜|µ−(N−2γ)(p−1) dx˜. (εN(p−1)−2pγ + εµ−2γ+N)‖φ‖∗ρ(x)−(N−2γ).Combining the above estimates, one hasI2 ≤C(εN(p−1)−2pγ + εµ−2γ+N)ρ(x)−(N−2γ)‖φ‖∗. (εN(p−1)−2pγ + εµ−2γ+N)ρ(x)ν‖φ‖∗,and thussup{dist(x,Σ)>σ}{ρ(x)−ν |φ |} ≤C(‖h‖∗∗+o(1)‖φ‖∗),182which implies, because our initial assumptions on φ , that there exists qi such thatsup{|x−qi|<σ}|x−qi|−µ |φ | ≥ 12 . (3.140)In the second step we study the region |x−qi|< σ . Without loss of generality,assume qi = 0. Recall that we are writing φ = I1+ I2. On the one hand,I1 =∫RNG(x, x˜)h(x˜)dx˜=∫{|x˜|>2σ}· · ·+∫{|x˜|< |x|2 }· · ·+∫{ |x|2 <|x˜|<2|x|}· · ·+∫{2|x|<|x˜|<2σ}· · ·≤ c‖h‖∗∗[∫{|x˜|>2σ}|x− x˜|−(N−2γ)|x˜|ν−2γ dx˜+∫{|x˜|< |x|2 }|x− x˜|−(N−2γ)|x˜|µ−2γ dx˜+∫{ |x|2 <|x˜|<2|x|}|x− x˜|−(N−2γ)|x˜|µ−2γ dx˜+∫{2|x|<|x˜|<2σ}|x− x˜|−(N−2γ)|x˜|µ−2γ dx˜]≤C‖h‖∗∗|x|µ .On the other hand, for I2, recall that φ is a solution to(−∆RN )γφ − pAN,p,γ u¯p−1ε φ = h.Define φ¯(x˜) = ε−µφ(ε x˜), then φ¯ satisfies(−∆RN )γ φ¯ − pAN,p,γup−11 φ¯ = ε2γ−µh(ε x˜).By the assumption that ‖h‖∗∗ → 0, one has that the right hand side tends to 0as j→ ∞. Since |φ¯(x˜)| ≤ C‖φ‖∗|x˜|µ locally uniformly, and by regularity theory,φ¯ ∈ C ηloc(RN \ {0}) for some η ∈ (0,1), thus passing to a subsequence, φ¯ → φ∞locally uniformly in any compact set, where φ∞ ∈ C α+2γµ,µ (RN \ {0}) is a solutionof(−∆RN )γφ∞− pAN,p,γup−11 φ∞ = 0 in RN \{0} (3.141)183(to handle the non-locality we may pass to the extension in a standard way). Sinceµ ≤ 0, it satisfies the condition in Proposition 3.7.2, from which we get that φ∞≡ 0,so φ¯ → 0.Now we go back to the calculation of I2. Here we use the change of variablex = εx1.I2 =∫{|x˜|<σ}p|x− x˜|−(N−2γ)u¯p−1ε φ dx˜ = εµ∫{|x˜|< σε }|x1− x˜|−(N−2γ)up−11 (x˜)φ¯(x˜)dx˜= εµ[∫{|x˜|< 1R }· · ·+∫{ 1R<|x˜|<R}· · ·+∫{R<|x˜|< σε }· · ·]=: J1+ J2+ J3,for some positive constant R large enough to be determined later. For J1, fix x,when ε → 0 one hasJ1 = εµ∫{|x˜|< 1R}|x1− x˜|−(N−2γ)up−11 (x˜)φ¯(x˜)dx˜≤ εµ‖φ‖∗|x1|−(N−2γ)∫{|x˜|< 1R }|x˜|µ−2γ dx˜≤CR−(N−2γ+µ)‖φ‖∗|x|µ .For J2 we use the fact that in this region φ¯ → 0, soJ2 = εµ∫{ 1R<|x˜|<R}|x1− x˜|−(N−2γ)up−11 (x˜)φ¯(x˜)dx˜= o(1)εµ∫{ 1R<|x˜|<R}1|x1− x˜|N−2γ dx˜ = o(1)εµ |x1|−(N−2γ) = o(1)|x|µ ,and finally,J3 = εµ∫{R<|x˜|< σε }|x1− x˜|−(N−2γ)up−11 (x˜)φ¯(x˜)dx˜= εµ‖φ‖∗[∫{R<|x˜|< |x1 |2 }|x1− x˜|−(N−2γ)|x˜|µup−11 dx˜+∫{ |x1 |2 <|x˜|<2|x1|}|x1− x˜|−(N−2γ)up−11 |x˜|µ dx˜+∫{2|x1|<|x˜|< σε }|x1− x˜|−(N−2γ)up−11 |x˜|µ dx˜]≤Cεµ |x1|µ‖φ‖∗|x1|−τ ≤ o(1)‖φ‖∗|x|µ184for some τ > 0.Combining all the above estimates, one has||x|−µ I1| ≤ o(1)(‖φ‖∗+1),which implies‖φ‖∗ ≤ o(1)‖φ‖∗+o(1)+‖h‖∗∗ = o(1).This is a contradiction with (3.140).For the more general case Rn \ Σ when Σ is a smooth k-dimensional sub-manifold, the argument is similar as above, the only difference is that one arrivesto the analogous to (3.141) in the estimate for I2 near Σ:(−∆Rn)γφ∞− pup−11 φ∞ = 0 in Rn \Rk.After the obvious rescaling by the constant AN,p,γ , where N = n− k, one uses Re-mark 3.5.8 and the injectivity result in Proposition 3.7.3 instead of the one in Pro-position 3.7.2. This completes the proof of Lemma 3.7.4.3.8 Fredholm properties - surjectivityOur analysis here follows closely the one in [142] for the local case. These lecturenotes are available online but, unfortunately, yet to be published.For the rest of the chapter, we will take the pair of dual weights µ, µ˜ such thatµ+ µ˜ =−(N−2γ) and ν+ ν˜ =−(n−2γ) satisfying− 2γp−1 < µ˜ < Re(γ−0 )≤−N−2γ2≤ Re(γ−0 )< µ < 0,− (n−2γ)< ν˜ < 0.(3.142)In order to consider the invertibility of the linear operators (3.122) and (3.123),defined in the spacesLε : C2γ+αµ˜,ν˜ → C 0,αµ˜−2γ,ν˜−2γ ,185it is simpler to consider the conjugate operatorL˜ε(w) := f−11 Lε( f2w), L˜ε : C2α+γµ˜+N−2γ2 ,ν˜+n−2γ2→ C 2α+γµ˜+N−2γ2 ,ν˜+n−2γ2, (3.143)where f2 is a weight ρ−n−2γ2 near infinity, and ρ−N−2γ2 near the singular set Σ, whilef1 is ρ−n+2γ2 near infinity, and ρ−N+2γ2 near the singular set. Recall that ρ is definedin Section 3.5.1. This conjugate operator is better behaved in weighted Hilbertspaces and simplifies the notation in the proof of Fredholm properties.3.8.1 Fredholm propertiesFredholm properties for extension problems related to this type of operators wereconsidered in [130, 131].In the notation of Section 3.5.1, and following the paper [132], we define theweighted Lebesgue space L2δ ,ϑ (Rn \ Σ). These are L2loc functions for which thenorm‖φ‖2L2δ ,ϑ (Rn\Σ) =∫Rn\BR|φ |2ρ−2γ−2ϑ dz+∫BR\Tσ|φ |2 dz+∫Tσ|φ |2ρN−1−2γ−2δ drdydθ (3.144)is finite. Here drdydθ denotes the corresponding measure in Fermi coordinatesr > 0, y ∈ Σ, θ ∈ SN−1. One defines accordingly, for γ > 0, weighted Sobolevspaces W 2γ,2δ ,ϑ with respect to the vector fields from Remark 3.5.3 (see [130] for theprecise definitions).The seemingly unusual normalization in the integrals in (3.144) is explainedby the change of variable w = f2φ . Indeed,‖w‖2L2δ ,ϑ =∫ − logR−∞∫Sn−1|w|2e2ϑ t dt˜dθ˜dr+∫{dist(·,Σ)>σ ,|z|<R}|w|2 dz+∫ +∞− logσ∫Σ∫SN−1|w|2e2δ t dtdydθ .We have186Lemma 3.8.1. For the choice of parameters−δ < µ˜+ N−2γ2 , −ϑ > ν˜+ n−2γ2 ,we have the continuous inclusionsC 2γ+αµ˜,ν˜ (Rn \Σ) ↪→ L2−δ ,−ϑ (Rn \Σ).The spaces L2δ ,ϑ and L2−δ ,−ϑ are dual with respect to the natural pairing〈φ1,φ2〉∗ =∫Rnφ1φ2,for φ1 ∈ L2δ ,ϑ , φ2 ∈ L2−δ ,−ϑ . Now, let L˜ε be the operator defined in (3.143). It is adensely defined, closed graph operator (this is a consequence of elliptic estimates).Then, relative to this pairing, the adjoint ofL˜ε : L2−δ ,−ϑ → L2−δ−2γ,−ϑ−2γ (3.145)is precisely(L˜ε)∗ = L˜ε : L2δ+2γ,ϑ+2γ → L2δ ,ϑ . (3.146)Now we fix µ , µ˜ , ν , ν˜ as in (3.142), and choose −δ < 0 slightly smaller thanµ˜+ N−2γ2 and −ϑ < 0 just slightly larger than ν˜+ n−2γ2 so that, in particular,− 2γp−1 + N−2γ2 <−δ < µ˜+ N−2γ2 < 0 < µ+ N−2γ2 < δ < N−2γ2 ,− n−2γ2 < ν˜+ n−2γ2 <−ϑ < 0 < ϑ < n−2γ2 ,(3.147)and we have the inclusions from Lemma 3.8.1. In addition, we can choose δ ,ϑ dif-ferent from the corresponding indicial roots. Higher order regularity is guaranteedby the results in Section 3.6.2. We will show:Proposition 3.8.2. Let δ ∈ (−N+2γ2 − 2γ, N−2γ2 ) and ϑ ∈ (−n−2γ2 , n−2γ2 ) be realnumbers satisfying (3.147).Assume that w ∈ L2δ ,ϑ is a solution toL˜εw = h˜ on Rn \Σ187for h˜ ∈ L2δ ,ϑ . Then we have the a priori estimate‖w‖L2δ ,ϑ . ‖h˜‖L2δ ,ϑ +‖w‖L2(K ), (3.148)whereK is a compact set in Rn \Σ. Translating back to the original operator Lε ,if φ is a solution to Lεφ = h in R\Σ, then (3.148) is rewritten as‖φ‖L2δ+2γ,ϑ+2γ . ‖h‖L2δ ,ϑ +‖φ‖L2(K ). (3.149)As a consequence, Lε has good Fredholm properties. The same is true for thelinear operators from (3.145) and (3.146).Proof. The proof here goes by subtracting suitable parametrices near Σ and nearinfinity thanks to Theorem 3.6.1. Then the remainder is a compact operator. Forsimplicity we set ε = 1.We first consider the point singularity case, i.e., k = 0, n = N.Step 1: (Localization) Let us study how the operator L1 : L2δ+2γ,ϑ+2γ → L2δ ,ϑ isaffected by localization, so that it is enough to work with functions supported nearinfinity and near the singular set.In the first step, assume that the singularity happens only at r = ∞ (but not atr = 0). We would like to patch a suitable parametrix at r = ∞. Let χ be a cut-offfunction such that χ = 1 in RN \BR, χ = 0 in BR/2. LetK = B2R, and seth1 := L1(χφ) = χL1φ +[L1,χ]φ ,where [·, ·] denotes the commutator operator. Contrary to the local case, the com-mutator term does not have compact support, but can still give good estimates inweighted Lebesgue spaces by carefully controlling the the tail terms. LetI(x) := [L1,χ]φ(x) = (−∆RN )γ(χφ)(x)−χ(x)(−∆RN )γφ(x)= kN,γ∫RNχ(x)−χ(x˜)|x− x˜|N+2γ φ(x˜)dx˜.Let us bound this integral in L2δ ,ϑ .188We first consider the case |x|  1. Note thatI(x) =∫B2R\BR/2χ(x)−χ(x˜)|x− x˜|N+2γ φ(x˜)dx˜+∫RN\B2Rφ(x˜)|x− x˜|N+2γ dx˜≤CR 2ϑ+2γ−N2 ‖φ‖L2δ+2γ,ϑ+2γ +‖φ‖L2(K ).We have that ‖I‖L2δ (B1) can be bounded by o(1)‖φ‖L2δ+2γ,ϑ+2γ + ‖φ‖L2(K ) for largeR if ϑ < N−2γ2 and δ <N−2γ2 .Next, for |x| ≥ 2R, we need to add the weight at infinity and calculate‖I‖L2ϑ (RN\B2R). ButI(x) =∫BR/2φ(x˜)|x− x˜|N+2γ dx˜+∫B2R\BR/2χ(x)−χ(x˜)|x− x˜|N+2γ φ(x˜)dx˜≤C‖φ‖L2δ+2γ,ϑ+2γR2ϑ+6γ+N2 |x|−(N+2γ) if δ >−N+2γ2 −2γ.One has that ‖I‖L2ϑ (RN\BR) = o(1)‖φ‖L2δ+2γ,ϑ+2γ if ϑ >−N+2γ2 −2γ .Now let 1≤ |x|< 2R, and calculate ‖I‖L2(B2R\B1). Again, we splitI(x) =∫BR/2χ(x)φ(x˜)|x− x˜|N+2γ dx˜+∫B2R\BR/2χ(x)−χ(x˜)|x− x˜|N+2γ φ(x˜)dx˜+∫RN\B2R(χ(x)−1)φ(x˜)|x− x˜|N+2γ dx˜=: I21+ I22+ I23.Similar to the above estimates, we can get that the L2 norm can be bounded by‖φ‖L2(K )+o(1)‖φ‖L2δ+2γ,ϑ+2γif ϑ < N−2γ2 . Thus we have shown that‖h1‖L2δ ,ϑ . ‖h‖L2δ ,ϑ +‖φ‖L2(K )+o(1)‖φ‖L2δ+2γ,ϑ+2γ ,so localization does not worsen estimate (3.149).189In addition, the localization at r = 0 is similar. One just needs to interchangethe role of r and 1/r, and ϑ by δ . Similar to the estimates in Step 5 below, onecan get that the error caused by the localization can be bounded by ‖φ‖L2(K )+o(1)‖φ‖L2δ+2γ,ϑ+2γ if −N+2γ2 −2γ < ϑ < N−2γ2 , −N+2γ2 −2γ < δ < N−2γ2 .Step 2: (The model operator) After localization around one of the singularpoints, say q1 = 0, the operator L1 can be approximated by the model operatorL1from (3.124), or by its conjugate given in (3.126). Moreover, recalling the notation(3.127) for the potential term and its asymptotics (3.128), it is enough to show that‖w‖L2δ . ‖h˜‖L2δ , (3.150)if w = w(t,θ) is a solution ofPg0γ w−κw = h˜, t ∈ R, θ ∈ SN−1, (3.151)that has compact support in t ∈ (0,∞). Here we have denoted κ = pAN,p,γ .Now project over spherical harmonics, so that w = ∑m wm(t)Em(θ), and wmsatisfiesPmγ wm−κwm = hm, t ∈ R.Our choice of weights (3.147) implies that there are no additional solutions tothe homogeneous problem and that we can simply write our solution as (3.96), inFourier variables. Then∫Re2δ t |wm(t)|2 dt =∫R|wm(ξ +δ i)|2 dξ =∫R1|Θmγ (ξ +δ i)−κ|2|hˆm(ξ +δ i)|2 dξ≤C∫R|hˆm(ξ +δ i)|2 dξ =∫Re2δ t |hm(t)|2 dt,(3.152)where we have used (3.106). (note that there are no poles on the R+ δ i line).Estimate (3.150) follows after taking sum in m and the fact that {Em} is an or-thonormal basis.For the estimate near infinity, we proceed in a similar manner, just approxim-ating the potential by τ = 0.190Step 3: (Compactness) Let {w j} be a sequence of solutions to L˜1w j = h˜ j withh˜ j ∈ L2δ ,ϑ . Assume that we have the uniform bound ‖w j‖L2δ ,ϑ ≤C. Then there existsa subsequence, still denoted by {w j}, that is convergent in L2δ ,ϑ norm. Indeed, bythe regularity properties of the equation, ‖w j‖W 2γ,2δ ,ϑ ≤C, which in particular, impliesa uniform W 2γ,2 in every compact setK . But this is enough to conclude that {w j}has a convergent subsequence in W 2γ,2(K ). Finally, estimate (3.148) implies thatthis convergence is also true in L2δ ,ϑ , as we claimed.Step 4: (Fredholm properties for L˜1) This is a rather standard argument. First,assume that the kernel is infinite dimensional, and take an orthonormal basis {w j}for this kernel. Then, by the claim in Step 3, we can find a Cauchy subsequence.But, for this,‖w j−w j′‖2 = ‖w j‖2+‖w j′‖2 = 2,a contradiction.Second, we show that the operator has closed range. Let {w j}, {h˜ j} be twosequences such thatL˜1w j = h˜ j and h˜ j→ h in L2δ ,ϑ . (3.153)Since Ker L˜1 is closed, we can use the projection theorem to write w j = w0j +w1jfor w0j ∈ Ker L˜1 and w1j ∈ (Ker L˜1)⊥. We have that L˜1w1j = h˜ j.Now we claim that this sequence is uniformly bounded, i.e., ‖w1j‖L2δ ,ϑ ≤C forevery j. By contradiction, assume that ‖w1j‖L2δ ,ϑ → ∞ as j→ ∞, and rescalew˜ j =w1j‖w1j‖L2δ ,ϑso that the new sequence has norm one in L2δ ,ϑ . From the previous remark, there isa convergent subsequence, still denoted by {w˜ j}, i.e., w˜ j → w˜ in L2δ ,ϑ . Moreover,we know that L˜1w˜ = 0. However, by assumption we have that w1j ∈ (Ker L˜1)⊥,therefore so does w˜. We conclude that w˜ must vanish identically, which is a con-tradiction with the fact that ‖w1j‖L2δ ,ϑ = 1. The claim is proved.191Now, using the remark in Step 3 again, we know that there exists a convergentsubsequence w1j → w1 in L2δ ,ϑ . This w1 must be regular, so we can pass to the limitin (3.153) to conclude that L˜1(w1) = h, as desired.Step 5. Now we consider the case that Σ is a sub-manifold of dimension k, andstudy the localization near a point in z0 ∈ Σ. In Fermi coordinates z = (x,y), this isa similar estimate to that of (3.92).First let χ be a cut-off function such that χ(r)= 1 for r≤ d and χ(r)= 0 for r≥2d. Define χ˜(z) = χ(dist(z,Σ)), and consider φ˜ = χ˜φ . Using Fermi coordinatesnear Σ and around a point z0 ∈ Σ, that can be taken to be z0 = (0,0) without loss ofgenerality, then, for z= (x,y) satisfying |x|  1, |y|  1, by checking the estimatesin the proof of Lemma 3.5.7, one can get that(−∆z)γ φ˜(z) = (1+ |x| 12 )(−∆Rn\Rk)γ φ˜(x,y)+ |x|−γ φ˜ +R2 (3.154)whereR2 =∫Σ∫{|x|β<|x˜|<2d}φ˜|x˜|N+2γ dx˜dy+ |x|−β (N+2γ)∫Σ∫{|x˜|<|x|β }φ˜ dx˜dy+ |x|βk∫Σ∫{|x|β<|x˜|<2d}φ˜|x˜|n+2γ dx˜dy≤ ‖φ˜‖L2δ+2γ |x|− 14 (N−2γ−2δ ),where we have used Ho¨lder inequality and that β = 12 . Here ‖φ˜‖L2δ+2γ is theweighted norm near Σ. One can easily check that the L2δ norm of R2 and |x|−γ φ˜are bounded by o(1)‖φ‖Lδ+2γ,ϑ+2γ for small d if δ < N−2γ2 .Next we consider the effect of the localization. LetI1(z) = [L1, χ˜]φ = kn,γ∫Rnχ˜(z)− χ˜(z˜)|z− z˜|n+2γ φ(z˜)dz˜.For |z|  1, one hasI1(z). |z|−(n+2γ)∫T2dφ(z˜)dz˜. dδ+N2 +3γ‖φ‖L2δ+2γ,ϑ+2γ |z|−(n+2γ).192Adding the weight at infinity we get that ‖I1‖L2ϑ (Rn\BR) can be bounded byo(1)‖φ‖L2δ+2γ,ϑ+2γ for d small and R large whenever δ > −N+2γ2 − 2γ, ϑ >−n+2γ2 −2γ .For |x|  1, one hasI1(z).∫T2d\Tdφ|z− z˜|n+2γ dz˜+∫T c2dφ|z− z˜|n+2γ dz˜.One can check that the L2δ term can be bounded by‖I1‖L2δ ≤C[‖φ‖L2(K )+R−n+2γ+2ϑ2 ‖φ‖L2ϑ+2γ (BcR)]≤C[‖φ‖L2(K )+o(1)‖φ‖L2δ+2γ,ϑ+2γ ]for d small and R large if δ < N−2γ2 , ϑ <n−2γ2 .For z ∈K , the estimate goes similarly for δ >−N+2γ2 −2γ . In conclusion, wehave‖I1‖L2δ ,ϑ ≤C[‖φ‖L2(K )+o(1)‖φ‖L2δ+2γ,ϑ+2γ ]. (3.155)Note that this estimate only uses the values of the function φ when |y| 1. Indeed,by checking the arguments in Lemma 5.7, the main term of the expansion for thefractional Laplacian in (3.154) comes from I11, i.e. for |y|  1. The contributionwhen |y| > |x|β is included in the remainder term |x|γ φ˜ +R2. The localizationaround the point z0 = (0,0) is now complete.Step 6. Next, after localization, we can replace (3.151) byPgkγ w− τw = h˜, in SN−1×Hk+1,and w is supported only near a point z0 ∈ ∂Hk+1, that can be taken arbitrarily. Wefirst consider the spherical harmonic decomposition for SN−1 and recall the symbolfor each projection from Theorem 3.3.5.The L2δ estimate follows similarly as in the case of points, but one uses theFourier-Helgason transform on hyperbolic space instead of the usual Fourier trans-form as in Theorem 3.3.5. Note, however, that the hyperbolic metric in (3.45) iswritten in half-space coordinates as dr2+|dy|2r2 = dt2 + e2t |dy|2, so in order to ac-193count for a weight of the form rδ one would need to use this transform writtenin rectangular coordinates. This is well known and comes Kontorovich-Lebedevformulas ([169]). Nevertheless, for our purposes it is more suitable to use thistransform in geodesic polar coordinates as it is described in Section 3.11. To ac-count for the weight, we just recall the following relation between two differentmodels for hyperbolic space Hk+1, the half space model with metrics dr2+|dy|2r2 andthe hyperboloid model with metric ds2+ sinhsgSk in geodesic polar coordinates:coshs = 1+|y|2+(r−2)24r.Since we are working locally near a point z0 ∈ ∂Hk+1, we can choose y = 0 in thisrelation, which yields that e−δ t = rδ = 2δ e−δ s. Thus we can use a weight of theform e−δ s in replacement for e−δ t .One could redo the theory of Section 3.6 using the Fourier-Helgason transforminstead. Indeed, after projection over spherical harmonics, and following (3.174),we can write for ζ ∈Hk+1,wm(ζ ) =∫Hk+1G (ζ ,ζ ′)h¯(ζ ′)dζ ′,where the Green’s function is given byG (ζ ,ζ ′) =∫ +∞−∞1Θmγ (λ )− τkλ (ζ ,ζ ′)dλ .The poles of 1Θmγ (λ )−τ are well characterized; in fact, they coincide with those in thepoint singularity case.But instead, we can take one further reduction and consider the projection overspherical harmonics in Sk. That is, in geodesic polar coordinates ζ = (s,ς), s > 0,ς ∈ Sk, we can write wm(s,ς) = ∑ j wm, j(s)E(k)j (ς), where E(k)j are the eigenfunc-tions for −∆Sk . Moreover, note that the symbol (3.48) is radial, so it commuteswith this additional projection.Now we can redo the estimate (3.152), just by taking into account the followingfacts: first, one also has a simple Plancherel formula (3.171). Second, for a radiallysymmetric function, the Fourier-Helgason transform takes the form of a simple194spherical transform (3.172). Third, the spherical function Φλ satisfies (3.173) andwe are taking a weight of the form eδ s. Finally, the expression for the symbol(3.48) is the same as in the point singularity case (3.38).This yields estimate (3.148) from which Fredholm properties follow immedi-ately.Remark 3.8.3. We do not claim that our restrictions on δ ,ϑ in Proposition 3.8.2are the sharpest possible (indeed, we chose them in the injectivity region for sim-plicity), but these are enough for our purposes.Gathering all restrictions on the weights we obtain:Corollary 3.8.4. The operator in (3.146) is injective, both inRN \{q1, . . . ,qK} andRn \⋃Σi. As a consequence, its adjoint (L˜∗ε)∗ = L˜ε given in (3.145) is surjective.Proof. Lemma 3.7.4 shows that, after performing the conjugation, L˜ε is injectivein C 2γ+αµ˜+N−2γ2 ,ν˜+n−2γ2. By regularity estimates and our choice of δ ,ϑ from (3.147), weimmediately obtain injectivity for (3.146). Since, thanks to the Fredholm proper-ties,Ker(L˜∗ε)⊥ = Rg(L˜ε),the Corollary follows.3.8.2 Uniform estimatesNow we return to the operator Lε defined in (3.123), the adjoint ofLε : L2−δ ,−ϑ → L2−δ−2γ,−ϑ−2γis justL∗ε : L2δ+2γ,ϑ+2γ → L2δ ,ϑ .From the above results, one knows that L∗ε is injective and Lε is surjective.Fixing the isomorphismspi2δ ,2ϑ : L2−δ ,−ϑ → L2δ ,ϑ ,195we may identify the adjoint L∗ε asL∗ε = pi−2δ ,−2ϑ ◦Lε ◦pi2δ ,2ϑ : L2−δ+2γ,−ϑ+2γ → L2−δ ,−ϑ .Now we have a new operatorLε = Lε ◦L∗ε : Lε ◦pi−2δ ,−2ϑ ◦Lε ◦pi2δ ,2ϑ : L2−δ+2γ,−ϑ+2γ → L2−δ−2γ,−ϑ−2γ .This map is an isomorphism. Hence there exists a bounded two sided inverseGε : L2−δ−2γ,−ϑ−2γ → L2−δ+2γ,−ϑ+2γ .Moreover, Gε = L∗ε ◦Gε is right inverse of Lε which map into the range of L∗ε . Wewill fix our inverse to be this one.From Corollary 3.8.4Gε : C 0,αµ˜−2γ,ν˜−2γ → C 4γ+αµ˜+2γ,ν˜+2γandGε : C0,αµ˜−2γ,ν˜−2γ → C 2γ+αµ˜,ν˜are bounded.We are now in the position to prove uniform surjectivity. It is a consequence ofthe following two results:Lemma 3.8.5. If u∈C 2γ+αµ˜,ν˜ and v∈C 4γ+αµ˜+2γ,ν˜+2γ solve equations Lεu= 0, u= L∗εv,then u≡ v≡ 0.Proof. Suppose u,v satisfy the given system, then one has LεL∗εv = 0. Considerw˜ = pi2δ ,2ϑv. Multiply the equation by w; integration by parts in Rn yields0 =∫wLε ◦pi−2δ ,−2ϑ ◦Lεw =∫pi−2δ ,−2ϑ |Lεw|2.Thus Lεw= 0. Moreover, since v∈C 4γ+αµ˜+2γ,ν˜+2γ , one has w∈C 2γ+αµ˜+2γ+2δµ˜ ,ν˜+2γ+2δν˜ ↪→C 2γ+αµ ′,ν ′ for some µ′ >Re(γ+0 ),ν′ >−(n−2γ) , thus by the injectivity property, onehas w≡ 0. We conclude then that u≡ v≡ 0.196Lemma 3.8.6. Let Gε be the bounded inverse of Lε introduced above, then forε small, Gε is uniformly bounded, i.e. for h ∈ C 0,αµ˜−2γ,ν˜−2γ , if u ∈ C 2γ+αµ˜,ν˜ ,v ∈C 4γ+αµ˜+2γ,ν˜+2γ solve the system Lεu = h and L∗εv = u, then one has‖u‖C 2γ+αµ˜,ν˜ (Rn\Σ)≤C‖h‖C 0,αµ˜−2γ,ν˜−2γ (Rn\Σ)for some C > 0 independent of ε small.Proof. The proof is similar to the proof of Lemma 3.7.4. So we just sketch theproof here and point out the differences. It is by contradiction argument. Assumethat there exists {ε(n)}→ 0 and a sequence of functions {h(n)} and solutions {u(n)},{v(n)} such that‖u‖C 2γ+αµ˜,ν˜ (Rn\Σ)= 1, ‖h‖C 0,αµ˜−2γ,ν˜−2γ (Rn\Σ)→ 0,and solve the equationLεu = h, L∗εv = u.Here note that, for simplicity, we have dropped the superindex (n). Then using theGreen’s representation formula, following the argument in Proposition 3.7.4, onecan show thatsup{dist(x,Σ)>σ}{ρ(x)−ν˜ |u|} ≤C(‖h‖C 0,αµ˜−2γ,ν˜−2γ+o(1)‖u‖C 2γ+αµ˜,ν˜),which implies that there exists qi such thatsup{|x−qi|<σ}|x−qi|−µ˜ |u| ≥ 12 . (3.156)In the second step we study the region {|x−qi|< σ}. Without loss of general-ity, assume qi = 0. Define the rescaled function as u¯ = ε−µ˜u(εx) and similarly forv¯ and h¯. Similarly to the argument in 3.7.4, u¯ will tend to a limit u∞ that solves(−∆)γu∞− pAN,p,γup−11 u∞ = 0 in RN .197If we show that this limit vanishes identically, u∞ ≡ 0, then we will reach a contra-diction with (3.156).For this, we wish to show that v¯ also tends to a limit. If‖v‖C 4γ+αµ˜+2γ,ν˜+2γ≤C0‖u‖C 2γ+αµ˜,ν˜ , (3.157)then it is true that the limit exists. If not, we can use the same contradiction argu-ment to show that after some scaling, v¯ will tend to a limit v∞ ∈ C 4γ+αµ˜+2γ,µ˜+2γ whichsolvesL∗1v∞ = 0.This implies that v≡ 0. This will give a contradiction and yield that (3.157) holdsfor some constant C0.By the above analysis we arrive at the limit problem, in which u∞,v∞ solveL1u∞ = 0, L∗1v∞ = u∞ in RN .Thus L1L∗1v∞ = 0. Multiply the equation by v∞ and integrate, one has L∗1v∞ = 0,which implies that v∞ ≡ 0. So also u∞ ≡ 0. Then, following the argument inLemma 3.7.4, one can get a contradiction. So the uniform surjectivity holds for allε small.3.9 Conclusion of the proofIf φ is a solution to(−∆Rn)γ(u¯ε +φ) = |u¯ε +φ |p in Rn \Σ,we first show that u¯ε +φ is positive in Rn \Σ.Indeed, for z near Σ, there exists R > 0 such that if ρ(z)< Rε , thenc1ρ(z)−2γp−1 < u¯ε < c2ρ(z)−2γp−1for some c1,c2 > 0. Since φ ∈ C 2,αµ˜,ν˜ , we have |φ | ≤ cρ(z)µ˜ . But µ˜ > − 2γp−1 , soit follows that u¯ε +φ > 0 near Σ. Since u¯ε +φ → 0 as |z| → ∞, by the maximum198principle (see for example Lemma 4.13 of [36]), we see that u¯ε +φ > 0 in Rn \Σ,so it is a positive solution and it is singular at all points of Σ.Next we will prove the existence of such φ . For this, we take an additionalrestriction on ν˜−(n−2γ)< ν˜ <− 2γp−1 .3.9.1 Solution with isolated singularities (RN \{q1, . . . ,qK})We first treat the case where Σ is a finite number of points. Recall that equation(−∆RN )γ(u¯ε +φ) = AN,p,γ |u¯ε +φ |p in RN \{q1, . . . ,qK}is equivalent to the following:Lε(φ)+Qε(φ)+ fε = 0, (3.158)where fε is defined in (3.86), Lε is the linearized operator from (3.122) and Qεcontains the remaining higher order terms. Because of Lemma 3.8.6, it is possibleto construct a right inverse for Lε with norm bounded independently of ε . DefineF(φ) := Gε [−Qε(φ)− fε ], (3.159)then equation (3.158) is reduced toφ = F(φ).Our objective is to show that F(φ) is a contraction mapping fromB toB, whereB = {φ ∈ C 2γ+αµ˜,ν˜ (Rn \Σ) : ‖φ‖∗ ≤ βεN−2pγp−1 }for some large positive β .In this section, ‖ · ‖∗ is the C 2γ+αµ˜,ν˜ norm, and ‖ · ‖∗∗ is the C 0,αµ˜−2γ,ν˜−2γ normwhere µ˜, ν˜ are taken as in the surjectivity section.First we have the following lemma:199Lemma 3.9.1. We have that, independently of ε small,‖Qε(φ1)−Qε(φ2)‖∗∗ ≤ 12l0 ‖φ1−φ2‖∗for all φ1, φ2 ∈B, where l0 = sup‖Gε‖.Proof. The estimates here are similar to Lemma 9 in [133]. For completeness, wegive here the proof.With some abuse of notation, in the following paragraphs the notation ‖·‖∗ and‖ · ‖∗∗ will denote the weighted C 0 norms and not the weighted C α norms (for thesame weights) that was defined in (3.139).First we show that there exists τ > 0 such that for φ ∈B, we have|φ(x)| ≤ 14u¯ε(x) for all x ∈k⋃i=1B(qi,τ).Indeed, from the asymptotic behaviour of u1 in Proposition 3.2.1 we know thatc1|x|−2γp−1 < uεi(x)< c2|x|−2γp−1 if |x|< Rεi,c1εN− 2pγp−1i |x|−(N−2γ) < uεi(x)< c2εN− 2pγp−1i |x|−(N−2γ) if Rεi ≤ |x|< τ.The claim follows because φ ∈B implies that|φ(x)|< cβεN− 2pγp−1ρ(x)µ˜ .Next, since | φu¯ε | ≤ 14 in B(qi,τ), by Taylor’ expansion,|Qε(φ1)−Qε(φ2)| ≤ c|u¯ε |p−2(|φ1|+ |φ2|)|φ1−φ2|.Thus for x ∈ B(qi,τ), we haveρ(x)2γ−µ˜ |Qε(φ1)−Qε(φ2)| ≤ cρ(x)µ˜+2γp−1 (‖φ1‖∗+‖φ2‖∗)‖φ1−φ2‖∗≤ cτ µ˜+ 2γp−1βεN− 2γp−1 ‖φ1−φ2‖∗.200The coefficient in front can be taken as small as desired by choosing ε small. Out-side the union of the balls B(qi,τ) we use the estimatesu¯ε(x)≤ cεN−2pγp−1 |x|−(N−2γ) and |φ | ≤ cεN− 2pγp−1 |x|ν˜ ,where c depends on τ but not on ε nor φ .For ρ ≥ τ and |x| ≤ R, we can neglect all factors involving ρ(x), so|Qε(φ1)−Qε(φ2)| ≤ c(|u¯ε |p−1+ |φ |p−1)|φ1−φ2| ≤ cε(p−1)(N−2pγp−1 )|φ1−φ2|≤ cε p(N−2γ)−N‖φ1−φ2‖∗,for which the coefficient can be as small as desired since p > NN−2γ .Lastly, for |x| ≥ R, in this region u¯ε = 0, soρ(x)2γ−ν˜ |Qε(φ1)−Qε(φ2)| ≤ cρ(x)2γ−ν˜(φ p−11 +φ p−12 )|φ1−φ2|≤ cρ(x)2γ−ν˜+pν˜εN(p−1)−2pγ‖φ1−φ2‖∗,and here the coefficient can be also chosen as small as we wish because ν˜ <− 2γp−1implies that 2γ− ν˜+ pν˜ < 0.Combining all the above estimates, one has‖Qε(φ1)−Qε(φ2)‖∗∗ ≤ 12l0 ‖φ1−φ2‖∗as desired.Now we go back to the original definition of the norms ‖ · ‖∗, ‖ · ‖∗∗ from(3.139). For this, we need to estimate the Ho¨lder norm of Qε(φ1)−Qε(φ2). Firstin each B(qi,τ),∇Qε(φ) = p((u¯ε +φ)p−1− u¯p−1ε − (p−1)u¯p−1ε φ)∇u¯ε+ p((u¯ε +φ)p−1− u¯p−1ε )∇φ ,and similarly as before, we can get thatρ(x)2γ+1−µ˜ |∇(Qε(φ1)−Qε(φ2))| ≤ cεN−2pγp−1 ‖φ1−φ2‖∗.201Moreover, for τ < ρ(x)< R,|∇(Qε(φ1)−Qε(φ2))| ≤ cε p(N−2γ)−N‖φ1−φ2‖∗.Lastly, for ρ(x)> R,∇Qε(φ1−φ2) = pφ p−11 ∇(φ1−φ2)+ p∇φ2(φ p−11 −φ p−12 ),which yieldsρ−ν˜+2γ+1|∇Qε(φ1−φ2)|≤ ρ2γ+1−ν˜[(‖φ1‖∗+‖φ2‖∗)p−1ρ(p−1)(ν˜−2γ)‖φ1−φ2‖∗ρ ν˜−2γ−1+‖φ2‖∗ρ ν˜−2γ−1‖φ p−11 −φ p−12 ‖∗]≤ cεN− 2pγp−1 ‖φ1−φ2‖∗.This completes the desired estimate for Qε(φ1)−Qε(φ2) and concludes the proofof the lemma.Recall that ‖ fε‖∗∗ ≤C0εN−2pγp−1 for some C0 > 0 from (3.88). Then the lemmaabove gives an estimate for the map (3.159). Indeed,‖F(φ)‖∗ ≤ l0[‖Qε(φ)‖∗∗+‖ fε‖∗∗]≤ l0‖Qε(φ)‖∗∗+ l0C0εN−2pγp−1≤ 12‖φ‖∗+ l0C0εN−2pγp−1 ≤ βεN− 2pγp−1 ,and‖F(φ1)−F(φ2)‖∗ ≤ l0‖Qε(φ1)−Qε(φ2)‖∗∗ ≤ 12‖φ1−φ2‖∗if we choose β > 2l0C0. So F(φ) is a contraction mapping from B to B. Thisimplies the existence of a solution φ to (3.158).2023.9.2 The general case Rn \Σ, Σ a sub-manifold of dimension kFor the more general case, only minor changes need to be made in the above argu-ment. The most important one comes from Lemma 3.5.7 and it says that the weightparameter µ must now lie in the smaller interval:− 2γp−1 < µ˜ < min{γ− 2γp−1 , 12 − 2γp−1 ,Re(γ−0 )}. (3.160)In this case, we only need to replace the exponent N − 2pγp−1 in the above ar-gument by q = min{ (p−3)γp−1 − µ˜, 12 − γ + (p−3)γp−1 − µ˜}, then q > 0 if µ˜ is chosento satisfy (3.160). We get a solution to (3.158), and this concludes the proof ofTheorem 3.1.1.3.10 Some known results on special functionsLemma 3.10.1. [4, 166] Let z ∈ C. The hypergeometric function is defined for|z|< 1 by the power series2F1(a,b;c;z) =∞∑n=0(a)n(b)n(c)nznn!=Γ(c)Γ(a)Γ(b)∞∑n=0Γ(a+n)Γ(b+n)Γ(c+n)znn!.It is undefined (or infinite) if c equals a non-positive integer. Some properties arei. The hypergeometric function evaluated at z = 0 satisfies2F1(a+ j,b− j;c;0) = 1; j =±1,±2, ... (3.161)ii. If |arg(1− z)|< pi , then2F1(a,b;c;z) =Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) 2F1 (a,b;a+b− c+1;1− z)+(1− z)c−a−bΓ(c)Γ(a+b− c)Γ(a)Γ(b) 2F1(c−a,c−b;c−a−b+1;1− z).(3.162)203iii. The hypergeometric function is symmetric with respect to first and secondarguments, i.e2F1(a,b;c;z) = 2F1(b,a;c;z). (3.163)iv. Let m ∈ N. The m-derivative of the hypergeometric function is given bydmdzm[(1− z)a+m−1 2F1(a,b;c;z)]= (−1)m(a)m(c−b)m(c)m(1− z)a−1 2F1(a+m,b;c+m;z). (3.164)Lemma 3.10.2. [4, 166] Let z ∈ C. Some well known properties of the Gammafunction Γ(z) areΓ(z¯) = Γ(z), (3.165)Γ(z+1) = zΓ(z), (3.166)Γ(z)Γ(z+ 12)= 21−2z√pi Γ(2z). (3.167)It is a meromorphic function in z ∈ C and its residue at each poles is given byRes(Γ(z),− j) = (−1)nj!, j = 0,1, . . . . (3.168)Let ψ(z) denote the Digamma function defined byψ(z) =d lnΓ(z)dz=Γ′(z)Γ(z).This function has the expansionψ(z) = ψ(1)+∞∑l=0(1l+1− 1l+ z). (3.169)Let B(z1,z2) denote the Beta function defined byB(z1,z2) =Γ(z1)Γ(z2)Γ(z1+ z2).204If z2 is a fixed number and z1 > 0 is big enough, then this function behavesB(z1,z2)∼ Γ(z2)(z1)−z2 .3.11 A review of the Fourier-Helgason transform onHyperbolic spaceConsider hyperbolic spaceHk+1, parameterized with coordinates ζ . It can be writ-ten as a symmetric space of rank one as the quotient Hk+1 ≈ SO(1,k+1)SO(k+1) . Fouriertransform on hyperbolic space is a particular case of the Helgason-Fourier trans-form on symmetric spaces. Some standard references are [26, 119, 168]; we mostlyfollow the exposition of Chapter 8 in [102].Hyperbolic spaceHk+1 may be defined as the upper branch of a hyperboloid inRk+2 with the metric induced by the Lorentzian metric in Rk+2 given by −dζ 20 +dζ 21 + . . .+ dζ 2k+1, i.e., Hk+1 = {(ζ0, . . . ,ζk+1) ∈ Rk+2 : ζ 20 − ζ 21 − . . .− ζ 2k+1 =1, ζ0 > 0}, which in polar coordinates may be parameterized asHk+1 = {ζ ∈ Rk+2 : ζ = (coshs,ς sinhs), s≥ 0, ς ∈ Sk},with the metric gHk+1 = ds2 + sinh2 sgSk . Under these definitions the Laplace-Beltrami operator is given by∆Hk+1 = ∂ss+ kcoshssinhs∂s+1sinh2 s∆Sk ,and the volume element isdµζ = sinhk s dsdς .We denote by [·, ·] the internal product induced by the Lorentzian metric, i.e.,[ζ ,ζ ′] = ζ0ζ ′0−ζ1ζ ′1− . . .−ζk+1ζ ′k+1.205The hyperbolic distance between two arbitrary points is given by dist(ζ ,ζ ′) =cosh−1([ζ ,ζ ′]), and in the particular case that ζ = (coshs,ς sinhs), ζ ′ = O,dist(ζ ,O) = s.The unit sphere SN−1 is identified with the subset {ζ ∈ Rk+2 : [ζ ,ζ ] = 0,ζ0 = 1}via the map b(ς) = (1,ς) for ς ∈ Sk.Given λ ∈ R and ω ∈ Sk, let hλ ,ω(ζ ) be the generalized eigenfunctions of theLaplace-Beltrami operator. This is,∆Hk+1hλ ,ω =−(λ 2+ k24)hλ ,ω .These may be explicitly written ashλ ,ω(ζ ) = [ζ ,b(ω)]iλ−k2 = (coshs− sinhs〈ς ,ω〉)iλ− k2 , ζ ∈Hk+1.In analogy to the Euclidean space, the Fourier transform on Hk+1 is defined byuˆ(λ ,ω) =∫Hk+1u(ζ )hλ ,ω(ζ )dµζ .Moreover, the following inversion formula holds:u(ζ ) =∫ ∞−∞∫Skh¯λ ,ω(ζ )uˆ(λ ,ω)dω dλ|c(λ )|2 , (3.170)where c(λ ) is the Harish-Chandra coefficient:1|c(λ )|2 =12(2pi)k+1|Γ(iλ +( k2)|2|Γ(iλ )|2 .There is also a Plancherel formula:∫Hk+1|u(ζ )|2 dµζ =∫R×SN−1|uˆ(λ ,ω)|2 dω dλ|c(λ )|2 , (3.171)which implies that the Fourier transform extends to an isometry between the Hilbertspaces L2(Hk+1) and L2(R+×Sk, |c(λ )|−2dλ dω).206If u is a radial function, then uˆ is also radial, and the above formulas simplify.In this setting, it is customary to normalize the measure of Sk to one in order notto account for multiplicative constants. Thus one defines the spherical Fouriertransform asuˆ(λ ) =∫Hk+1u(ζ )Φ−λ (ζ )dµζ , (3.172)whereΦλ (ζ ) =∫Skh−λ ,ω(ζ )dωis known as the elementary spherical function. In addition, (3.170) reduces tou(ζ ) =∫ ∞−∞uˆ(λ )Φλ (ζ )dλ|c(λ )|2 .There are many explicit formulas for Φλ (ζ ) (we also write Φλ (s), since it is aradial function). In particular, Φ−λ (s) =Φλ (s) =Φλ (−s), which yields regularityat the origin s = 0. Here we are interested in its asymptotic behavior. Indeed,Φλ (s)∼ e(iλ−k2 )s as s→+∞. (3.173)It is also interesting to observe that∆̂Hk+1u =−(λ 2+ k24)uˆ.We define the convolution operator asu∗ v(ζ ) =∫Hk+1u(ζ ′)v(τ−1ζ ζ′)dµζ ′ ,where τζ :Hk+1→Hk+1 is an isometry that takes ζ into O. If v is a radial function,then the convolution may be written asu∗ v(ζ ) =∫Hk+1u(ζ ′)v(dist(ζ ,ζ ′))dµζ ′ ,and we have the propertyû∗ v = uˆ vˆ,in analogy to the usual Fourier transform.207On hyperbolic space there is a well developed theory of Fourier multipliers. InL2 spaces everything may be written out explicitly. For instance, let m(λ ) be a mul-tiplier in Fourier variables. A function uˆ(λ ,ω) = mˆ(λ )u0(λ ,ω), by the inversionformula for the Fourier transform (3.170) and expression (3.11), may be written asu(x) =∫ ∞−∞∫Skm(λ )uˆ0(λ ,ω)h¯λ ,ω(ζ )dω dλ|c(λ )|2=∫ ∞−∞∫Hk+1m(λ )u0(ζ ′)kλ (ζ ,ζ ′)dµζ ′ dλ ,(3.174)where we have denotedkλ (ζ ,ζ ′) =1|c(λ )|2∫Skh¯λ ,ω(ζ )hλ ,ω(ζ ′)dω.It is known that kλ is invariant under isometries, i.e.,kλ (ζ ,ζ ′) = kλ (τζ ,τζ ′),for all τ ∈ SO(1,k+1), and in particular,kλ (ζ ,ζ ′) = kλ (dist(ζ ,ζ ′)),so many times we will simply write kλ (ρ) for ρ = dist(ζ ,ζ ′). We recall the fol-lowing formulas for kλ :Lemma 3.11.1 ([102]). For k+1≥ 3 odd,kλ (ρ) = ck(∂ρsinhρ) k2(cosλρ),and for k+1≥ 2 even,kλ (ρ) = ck∫ ∞ρsinh ρ˜√cosh ρ˜− coshρ(∂ρ˜sinh ρ˜) k+12(cosλρ˜)dρ˜.208Chapter 4Extremals for HyperbolicHardy–Schro¨dinger Operators4.1 IntroductionHardy–Schro¨dinger operators on manifolds are of the form ∆g−V , where ∆g is theLaplace–Beltrami operator and V is a potential that has a quadratic singularity atsome point of the manifold. For hyperbolic spaces, Carron [46] showed that, justlike in the Euclidean case and with the same best constant, the following inequalityholds on any Cartan–Hadamard manifold M,(n−2)24∫Mu2dg(o,x)2dvg ≤∫M|∇gu|2 dvg for all u ∈C∞c (M),where dg(o,x) denotes the geodesic distance to a fixed point o∈M. There are manyother works identifying suitable Hardy potentials, their relationship with the ellipticoperator on hand, as well as corresponding energy inequalities [6, 57, 71, 123, 126,181? ]. In the Euclidean case, the Hardy potential V (x) = 1|x|2 is distinguished bythe fact that u2|x|2 has the same homogeneity as |∇u|2, but also u2∗(s)|x|s , where 2∗(s) =2(n−s)n−2 and 0 ≤ s < 2. In other words, the integrals∫Rnu2|x|2 dx,∫Rn|∇u|2 dx and∫Rnu2∗(s)|x|s dx are invariant under the scaling u(x) 7→ λn−22 u(λx), λ > 0, which makes209corresponding minimization problem non-compact, hence giving rise to interestingconcentration phenomena. In [5], Adimurthi and Sekar use the fundamental solu-tion of a general second order elliptic operator to generate natural candidates andderive Hardy-type inequalities. They also extended their arguments to Riemannianmanifolds using the fundamental solution of the p-Laplacian. In [71], Devyver,Fraas and Pinchover study the case of a general linear second order differential op-erator P on non-compact manifolds. They find a relation between positive super-solutions of the equation Pu= 0, Hardy-type inequalities involving P and a weightW , as well as some properties of the spectrum of a corresponding weighted oper-ator.In this paper, we shall focus on the Poincare´ ball model of the hyperbolic spaceBn, n ≥ 3, that is the Euclidean unit ball B1(0) := {x ∈ Rn : |x| < 1} endowedwith the metric gBn =(21−|x|2)2gEucl . This framework has the added feature ofradial symmetry, which plays an important role and contributes to the richnessof the structure. In this direction, Sandeep and Tintarev [153] recently came upwith several integral inequalities involving weights on Bn that are invariant underscaling, once restricted to the class of radial functions (see also Li and Wang [126]).As described below, this scaling is given in terms of the fundamental solution ofthe hyperbolic Laplacian ∆Bnu = divBn(∇Bnu). Indeed, letf (r) :=(1− r2)n−2rn−1and G(r) :=∫ 1rf (t)dt, (4.1)where r =√∑ni=1 x2i denotes the Euclidean distance of a point x ∈ B1(0) to theorigin. It is known that 1nωn−1 G(r) is a fundamental solution of the hyperbolicLaplacian ∆Bn . As usual, the Sobolev space H1(Bn) is defined as the completionof C∞c (Bn) with respect to the norm ‖u‖2H1(Bn) =∫Bn|∇Bnu|2dvgBn . We denote byH1r (Bn) the subspace of radially symmetric functions. For functions u ∈ H1r (Bn),we consider the scalinguλ (r) = λ−12 u(G−1(λG(r))), λ > 0. (4.2)210In [153], Sandeep–Tintarev have noted that for any u ∈H1r (Bn) and p≥ 1, one hasthe following invariance property:∫Bn|∇Bnuλ |2 dvgBn =∫Bn|∇Bnu|2 dvgBn and∫BnVp|uλ |p dvgBn =∫BnVp|u|p dvgBn ,whereVp(r) :=f (r)2(1− r2)24(n−2)2G(r) p+22. (4.3)In other words, the hyperbolic scaling r 7→ G−1(λG(r)) is quite analogous tothe Euclidean scaling. Indeed, in that case, by taking G(ρ) = ρ2−n, we see thatG−1(λG(ρ)) = λ = λ12−n for ρ = |x| in Rn. Also, note that G is –up to a constant–the fundamental solution of the Euclidean Laplacian ∆ in Rn. The weights Vp havethe following asymptotic behaviors, for n≥ 3 and p > 1,Vp(r) =c0(n, p)rn(1−p/2∗)(1+o(1)) as r→ 0,c1(n, p)(1− r)(n−1)(p−2)/2 (1+o(1)) as r→ 1.In particular, for n≥ 3, the weight V2(r) = 14(n−2)2(f (r)(1−r2)G(r))2 ∼r→0 14r2 , andat r = 1 has a finite positive value. In other words, the weight V2 is qualitativelysimilar to the Euclidean Hardy weight, and Sandeep–Tintarev have indeed estab-lished the following Hardy inequality on the hyperbolic space Bn (Theorem 3.4 of[153]). Also, see [71] where they deal with similar Hardy weights.(n−2)24∫BnV2|u|2 dvgBn ≤∫Bn|∇Bnu|2 dvgBn for any u ∈ H1(Bn).They also show in the same paper the following Sobolev inequality, i.e., for someconstant C > 0.(∫BnV2∗ |u|2∗dvgBn)2/2∗≤C∫Bn|∇Bnu|2 dvgBn for any u ∈ H1(Bn),211where 2∗= 2n(n−2) . By interpolating between these two inequalities taking 0≤ s≤ 2,one easily obtain the following Hardy–Sobolev inequality.Lemma 4.1.1. If γ < (n−2)24 , then there exists a constant C > 0 such that, for anyu ∈ H1(Bn),C(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s)≤∫Bn|∇Bnu|2 dvgBn − γ∫BnV2|u|2 dvgBn ,where 2∗(s) := 2(n−s)(n−2) .Note that, up to a positive constant, we have V2∗(s) ∼r→0 1rs , adding to the ana-logy with the Euclidean case, where we have for any u ∈ H1(Rn),C(∫Rn|u|2∗(s)|x|s dx)2/2∗(s)≤∫Rn|∇u|2 dx− γ∫Rn|u|2|x|2 dx.Motivated by the recent progress on the Euclidean Hardy–Schro¨dinger equation(See for example Ghoussoub–Robert [105, 106], and the references therein), weshall consider the problem of existence of extremals for the corresponding bestconstant, that isµγ,λ (Ω) := infu∈H10 (Ω)\{0}∫Ω|∇Bnu|2 dvgBn − γ∫ΩV2|u|2 dvgBn −λ∫Ω|u|2 dvgBn(∫ΩV2∗(s)|u|2∗(s) dvgBn)2/2∗(s) ,(4.4)where H10 (Ω) is the completion of C∞c (Ω) with respect to the norm ‖u‖ =√∫Ω|∇u|2 dvgBn . Similarly to the Euclidean case, and once restricted to radialfunctions, the general Hardy–Sobolev inequality for the hyperbolic Hardy–Schro¨dinger operator is invariant under hyperbolic scaling described in (4.2), Thisinvariance makes the corresponding variational problem non-compact and theproblem of existence of minimizers quite interesting.In Proposition 4.3.3, we start by showing that the extremals for the minimiza-tion problem (4.4) in the class of radial functions H1r (Bn) can be written explicitly212as:U(r) = c(G(r)−2−sn−2α−(γ)+G(r)−2−sn−2α+(γ))− n−22−s,where c is a positive constant and α±(γ) satisfyα±(γ) =12±√14− γ(n−2)2 .In other words, we show thatµ radγ,0 (Bn) := infu∈H1r (Bn)\{0}∫Bn|∇Bnu|2 dvgBn − γ∫BnV2|u|2 dvgBn(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s) (4.5)is attained by U .Note that the radial function Gα(r) is a solution of −∆Bnu− γV2u = 0 on Bn \{0} if and only if α = α±(γ). These solutions have the following asymptoticbehaviorG(r)α±(γ) ∼ c(n,γ)r−β±(γ) as r→ 0,whereβ±(γ) =n−22±√(n−2)24− γ.These then yield positive solutions to the equation−∆Bnu− γV2u =V2∗(s)u2∗(s)−1 in Bn.We point out the paper [128] (also see [22, 23, 101]), where the authors consideredthe counterpart of the Brezis–Nirenberg problem on Bn (n≥ 3), and discuss issuesof existence and non-existence for the equation−∆Bnu−λu = u2∗−1 in Bn,in the absence of a Hardy potential.213Next, we consider the attainability of µγ,λ (Ω) in subdomains of Bn withoutnecessarily any symmetry. In other words, we will search for positive solutions forthe equation −∆Bnu− γV2u−λu =V2∗(s)u2∗(s)−1 in Ωu≥ 0 in Ωu = 0 on ∂Ω,(4.6)where Ω is a compact smooth subdomain of Bn such that 0 ∈ Ω, but Ω does nottouch the boundary of Bn and λ ∈ R. Note that the metric is then smooth onsuch Ω, and the only singularity we will be dealing with will be coming from theHardy-type potential V2 and the Hardy–Sobolev weight V2∗(s), which behaves like1r2 (resp.,1rs ) at the origin. This is analogous to the Euclidean problem on boundeddomains considered by Ghoussoub–Robert [105, 106]. We shall therefore relyheavily on their work, at least in dimensions n ≥ 5. Actually, once we perform aconformal transformation, the equation above reduces to the study of the followingtype of problems on bounded domains in Rn:−∆v−(γ|x|2 +hγ,λ (x))v = b(x) v2∗(s)−1|x|s in Ωv≥ 0 in Ωv = 0 on ∂Ω,where b(x) is a positive C0(Ω) function withb(0) =(n−2) n−sn−222−s, (4.7)hγ,λ (x) = γa(x)+4λ −n(n−2)(1−|x|2)2 ,214a(x) = a(r) =4r +8+g3(r) when n = 3,8log 1r −4+g4(r) when n = 4,4(n−2)n−4 + rgn(r) when n≥ 5.(4.8)with gn(0) = 0, for all n≥ 3. Ghoussoub–Robert [106] have recently tackled suchan equation, but in the case where h(x) and b(x) are constants. We shall explorehere the extent of which their techniques could be extended to this setting. To startwith, the following regularity result will then follow immediately.Theorem 4.1.2 (Regularity). Let Ω b Bn, n ≥ 3, and γ < (n−2)24 . If u 6≡ 0 is anon-negative weak solution of the equation (4.6) in the hyperbolic Sobolev spaceH1(Ω), thenlim|x|→0u(x)G(|x|)α− = K > 0.We also need to define a notion of mass of a domain associated to the operator−∆Bn− γV2−λ . We therefore show the following.Theorem 4.1.3 (The hyperbolic Hardy-singular mass of Ωb Bn). Let 0 ∈Ωb Bn,n ≥ 3, and γ < (n−2)24 . Let λ ∈ R be such that the operator −∆Bn − γV2− λ iscoercive. Then, there exists a solution KΩ ∈C∞(Ω\{0}) to the linear problem,−∆BnKΩ− γV2KΩ−λKΩ = 0 in ΩKΩ ≥ 0 in ΩKΩ = 0 on ∂Ω,(4.9)such that KΩ(x)'|x|→0 c G(|x|)α+ for some positive constant c. Furthermore,1. If K′Ω ∈C∞(Ω\{0}) is another solution of the above linear equation, thenthere exists a C > 0 such that K′Ω =CKΩ.2. If γ > max{n(n−4)4 ,0}, then there exists mHγ,λ (Ω) ∈ R such thatKΩ(x) =G(|x|)α++mHγ,λ (Ω)G(|x|)α−+o(G(|x|)α−) as x→ 0. (4.10)215The constant mHγ,λ (Ω) will be referred to as the hyperbolic mass of the do-main Ω associated with the operator −∆Bn− γV2−λ .And just like the Euclidean case, solutions exist in high dimensions, whilethe positivity of the “hyperbolic mass”will be needed for low dimensions. Moreprecisely,Theorem 4.1.4. LetΩbBn (n≥ 3) be a smooth domain with 0∈Ω, 0≤ γ < (n−2)24and let λ ∈R be such that the operator −∆Bn− γV2−λ is coercive. Then, the bestconstant µγ,λ (Ω) is attained under the following conditions:1. n≥ 5, γ ≤ n(n−4)4and λ >n−2n−4(n(n−4)4− γ).2. n = 4, γ = 0 and λ > 2.3. n = 3, γ = 0 and λ >34.4. n≥ 3, max{n(n−4)4,0}< γ <(n−2)24and mHγ,λ (Ω)> 0.As mentioned above, the above theorem will be proved by using a conformal trans-formation that reduces the problem to the Euclidean case, already considered byGhoussoub–Robert [106]. Actually, this leads to the following variation of theproblem they considered, where the perturbation can be singular but not as muchas the Hardy potential.Theorem 4.1.5. Let Ω be a bounded smooth domain in Rn, n≥ 3, with 0 ∈Ω and0≤ γ < (n−2)24 . Let h ∈C1(Ω\{0}) be such thath(x) =−C1|x|−θ log |x|+ h˜(x) where limx→0|x|θ h˜(x) = C2, (4.11)for some 0≤ θ < 2 and C1,C2 ∈ R, and the operator −∆−(γ|x|2 +h(x))is coer-cive. Also, assume that b(x) is a non-negative function in C0(Ω) with b(0) > 0.216Then the best constantµγ,h(Ω) := infu∈H10 (Ω)\{0}∫Ω(|∇u|2−(γ|x|2 +h(x))u2)dx(∫Ωb(x)|u|2∗(s)|x|s dx)2/2∗(s) (4.12)is attained if one of the following two conditions is satisfied:1. γ ≤ (n−2)24 − (2−θ)24 and, either C1 > 0 or {C1 = 0, C2 > 0};2. (n−2)24 − (2−θ)24 < γ <(n−2)24 and mγ,h(Ω)> 0, where mγ,h(Ω) is the mass ofthe domain Ω associated to the operator −∆−(γ|x|2 +h(x)).The paper is organized as follows. In Section 2, we introduce the Hardy–Sobolev type inequalities in hyperbolic space. In Section 3, we find the explicitsolutions for Hardy–Sobolev equations corresponding to (4.5) on Bn. In section 4,we show that our main equation (4.6) can be transformed into the Hardy–Sobolevtype equations in Euclidean space under a conformal transformation. Section 5 isthen devoted to establish the existence results for (4.6) on compact submanifoldsof Bn by studying the transformed equations in Euclidean space.4.2 Hardy–Sobolev type inequalities in hyperbolic spaceThe starting point of the study of existence of weak solutions of the above problemsare the following inequalities which will guarantee that functionals (4.4) and (4.5)are well defined and bounded below on the right function spaces. The Sobolevinequality for hyperbolic space [153] asserts that for n≥ 3, there exists a constantC > 0 such that(∫BnV2∗ |u|2∗dvgBn)2/2∗≤C∫Bn|∇Bnu|2 dvgBn for all u ∈ H1(Bn),where 2∗ = 2nn−2 and V2∗ is defined in (4.3). The Hardy inequality on Bn [153]states:(n−2)24∫BnV2|u|2 dvgBn ≤∫Bn|∇Bnu|2 dvgBn for all u ∈ H1(Bn).217Moreover, just like the Euclidean case, (n−2)24 is the best Hardy constant in theabove inequality on Bn, i.e.,γH :=(n−2)24= infu∈H1(Bn)\{0}∫Bn|∇Bnu|2 dvgBn∫BnV2|u|2 dvgBn.By interpolating these inequalities via Ho¨lder’s inequality, one gets the followingHardy–Sobolev inequalities in hyperbolic space.Lemma 4.2.1. Let 2∗(s) = 2(n−s)n−2 where 0 ≤ s ≤ 2. Then, there exist a positiveconstant C such thatC(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s)≤∫Bn|∇Bnu|2 dvgBn (4.13)for all u ∈ H1(Bn). If γ < γH := (n−2)24 , then there exists Cγ > 0 such thatCγ(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s)≤∫Bn|∇Bnu|2 dvgBn − γ∫BnV2|u|2 dvgBn (4.14)for all u ∈ H1(Bn).Proof. Note that for s= 0 (resp., s= 2) the first inequality is just the Sobolev (resp.,the Hardy) inequality in hyperbolic space. We therefore have to only consider thecase where 0< s< 2 where 2∗(s)> 2. Note that 2∗(s) =( s2)2+(2− s2)2∗, andsoV2∗(s) =f (r)2(1− r)24(n−2)2G(r)(1√G(r))2∗(s)=(f (r)2(1− r)24(n−2)2G(r)) s2+2−s2(1√G(r))( s2 )2+( 2−s2 )2∗= f (r)2(1− r)24(n−2)2G(r)(1√G(r))2 s2  f (r)2(1− r)24(n−2)2G(r)(1√G(r))2∗ 2−s2=Vs22 V2−s22∗ .218Applying Ho¨lder’s inequality with conjugate exponents 2s and22−s , we obtain∫BnV2∗(s)|u|2∗(s) dvgBn =∫Bn(|u|2) s2Vs22 ·(|u|2∗) 2−s2V2−s22∗ dvgBn≤(∫BnV2|u|2 dvgBn) s2(∫BnV2∗ |u|2∗dvgBn) 2−s2≤C−1(∫Bn|∇Bnu|2 dvgBn) s2(∫Bn|∇Bnu|2 dvgBn) 2∗22−s2=C−1(∫Bn|∇Bnu|2 dvgBn) 2∗(s)2.It follows that for all u ∈ H1(Bn),∫Bn|∇Bnu|2 dvgBn − γ∫BnV2|u|2 dvgBn(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s) ≥(1− γγH) ∫Bn|∇Bnu|2 dvgBn(∫BnV2∗(s)|u|2∗(s) dvgBn)2/2∗(s) .Hence, (4.13) implies (4.14) whenever γ < γH := (n−2)24 .The best constant µγ(Bn) in inequality (4.14) can therefore be written as:µγ(Bn) = infu∈H1(Bn)\{0}∫Bn|∇Bnu|2 dvgBn − γ∫BnV2|u|2 dvgBn(∫BnV2∗(s)|u|2∗(s) dvBn)2/2∗(s) .Thus, any minimizer of µγ(Bn) satisfies –up to a Lagrange multiplier– the follow-ing Euler–Lagrange equation−∆Bnu− γV2u =V2∗(s)|u|2∗(s)−2u, (4.15)where 0≤ s < 2 and 2∗(s) = 2(n−s)n−2 .2194.3 The explicit solutions for Hardy–Sobolev equationson BnWe first find the fundamental solutions associated to the Hardy–Schro¨dinger oper-ator on Bn, that is the solutions for the equation −∆Bnu− γV2u = 0.Lemma 4.3.1. Assume γ < γH := (n−2)24 . The fundamental solutions of−∆Bnu− γV2u = 0are given byu±(r) = G(r)α±(γ) ∼(1n−2r2−n)α±(γ)as r→ 0,(2n−2n−1(1− r)n−1)α±(γ)as r→ 1,whereα±(γ) =β±(γ)n−2 and β±(γ) =n−22±√(n−2)24− γ. (4.16)Proof. We look for solutions of the form u(r) = G(r)−α . To this end we performa change of variable σ = G(r), v(σ) = u(r) to arrive at the Euler-type equation(n−2)2v′′(σ)+ γσ−2v(σ) = 0 in (0,∞).It is easy to see that the two solutions are given by v(σ)=σ±, or u(r)= c(n,γ)r−β±where α± and β± are as in (4.16).Remark 4.3.2. We point out that u±(r)∼ c(n,γ)r−β±(γ) as r→ 0.Proposition 4.3.3. Let −∞< γ < (n−2)24 . The equation−∆Bnu− γV2u =V2∗(s)u2∗(s)−1 in Bn, (4.17)220has a family of positive radial solutions which are given byU(G(r)) = c(G(r)−2−sn−2α−(γ)+G(r)−2−sn−2α+(γ))− n−22−s= c(G(r)− 2−s(n−2)2 β−(γ)+G(r)− 2−s(n−2)2 β+(γ))− n−22−s,where c is a positive constant and α±(γ) and β±(γ) satisfy (4.16).Proof. With the same change of variable σ = G(r) and v(σ) = u(r) we have(n−2)2v′′(σ)+ γσ−2v(σ)+σ− 2∗(s)+22 v2∗(s)−1(σ) = 0 in (0,∞).Now, set σ = τ2−n and w(τ) = v(σ)τ1−n(τn−1w′(τ))′+ γτ−2w(τ)+w(τ)2∗(s)−1 = 0 on (0,∞).The latter has an explicit solutionw(τ) = c(τ2−sn−2β−(γ)+ τ2−sn−2β+(γ))− n−22−s,where c is a positive constant. This translates to the explicit formulau(r) = c(G(r)−2−sn−2α−(γ)+G(r)−2−sn−2α+(γ))− n−22−s= c(G(r)− 2−s(n−2)2 β−(γ)+G(r)− 2−s(n−2)2 β+(γ))− n−22−s.Remark 4.3.4. We remark that, in the special case γ = 0 and s = 0, Sandeep–Tintarev [153] proved that the following minimization problemµ0(Bn) = infu∈H1r (Bn)\{0}∫Bn|∇Bnu|2 dvgBn∫BnV2∗ |u|2∗dvgBnis attained.221Remark 4.3.5. The change of variable σ = G(r) offers a nice way of viewing theradial aspect of hyperbolic space Bn in parallel to the one in Rn in the followingsense.• The scaling r 7→ G−1(λG(r)) for r = |x| in Bn corresponds to σ 7→ λσ in(0,∞), which in turn corresponds to ρ 7→ λρ = G−1(λG(ρ)) for ρ = |x| inRn, once we set G(ρ) = ρ2−n and λ = λ12−n ;• One has a similar correspondence with the scaling-invariant equations: if usolves−∆Bnu− γV2u =V2∗(s)u2∗(s)−1 in Bn,then1. as an ODE, and once we set v(σ) = u(r), σ = G(r), it is equivalent to−(n−2)2v′′(σ)− γσ−2v(σ) = σ− 2∗(s)+22 v(σ)2∗(s)−1 on (0,∞);(4.18)2. as a PDE on Rn, and by setting v(σ) = u(ρ), σ = G(ρ), it is in turnequivalent to−∆v− γ|x|2 v =1|x|s v2∗(s)−1 in Rn.This also confirm that the potentials V2∗(s) are the “correct” ones associatedto the power |x|−s.• The explicit solution u on Bn is related to the explicit solution w on Rn in thefollowing way:u(r) = w(G(r)−1n−2).222• Under the above setting, it is also easy to see the following integral identities:∫Bn|∇Bnu|2 dvgnB =∫ ∞0v′(σ)2 dσ∫BnV2u2 dvgnB =1(n−2)2∫ ∞0v2(σ)σ2dσ∫BnVpup dvgnB =1(n−2)2∫ ∞0vp(σ)σp+22dσ ,which, in the same way as above, equal to the corresponding Euclidean in-tegrals.4.4 The corresponding perturbed Hardy–Schro¨dingeroperator on Euclidean spaceWe shall see in the next section that after a conformal transformation, the equation(4.6) is transformed into the Euclidean equation−∆u−(γ|x|2 +h(x))u = b(x)u2∗(s)−1|x|s in Ω,u > 0 in Ω,u = 0 on ∂Ω,(4.19)where Ω is a bounded domain in Rn, n≥ 3, h ∈C1(Ω\{0}) with lim|x|→0|x|2h(x) = 0is such that the operator −∆−(γ|x|2 +h(x))is coercive and b(x) ∈C0(Ω) is non-negative with b(0) > 0. The equation (4.19) is the Euler–Lagrange equation forfollowing energy functional on D1,2(Ω),JΩγ,h(u) :=∫Ω(|∇u|2−(γ|x|2 +h(x))u2)dx( ∫Ωb(x)|u|2∗(s)|x|s dx)2/2∗(s) .223Here D1,2(Ω) – or H10 (Ω) if the domain is bounded – is the completion of C∞c (Ω)with respect to the norm given by ||u||2 = ∫Ω|∇u|2 dx. We letµγ,h(Ω) := infu∈D1,2(Ω)\{0}JΩγ,h(u)A standard approach to find minimizers is to compare µγ,h(Ω) with µγ,0(Rn). It isknow that µγ,0(Rn) is attained when γ ≥ 0, and minimizers are explicit and takethe formUε(x) := cγ,s(n) · ε− n−22 U( xε)= cγ,s(n) ·(ε2−sn−2 ·β+(γ)−β−(γ)2ε2−sn−2 ·(β+(γ)−β−(γ))|x| (2−s)β−(γ)n−2 + |x| (2−s)β+(γ)n−2) n−22−sfor x ∈ Rn \ {0}, where ε > 0, cγ,s(n) > 0, and β±(γ) are defined in (4.16), see[105]. In particular, there exists χ > 0 such that−∆Uε − γ|x|2Uε = χU2∗(s)−1ε|x|s in Rn \{0}. (4.20)We shall start by analyzing the singular solutions and then define the mass of adomain associated to the operator −∆−(γ|x|2 +h(x)).Proposition 4.4.1. Let Ω be a smooth bounded domain in Rn such that 0 ∈Ω andγ < (n−2)24 . Let h ∈C1(Ω \ {0}) be such that lim|x|→0 |x|τh(x) exists and is finite, forsome 0≤ τ < 2, and that the operator −∆− γ|x|2 −h(x) is coercive. Then1. There exists a solution K ∈C∞(Ω\{0}) for the linear problem−∆K−(γ|x|2 +h(x))K = 0 in Ω\{0}K > 0 in Ω\{0}K = 0 on ∂Ω,(4.21)224such that for some c > 0,K(x)'x→0 c|x|β+(γ) . (4.22)Moreover, if K′ ∈ C∞(Ω \ {0}) is another solution for the above equation,then there exists λ > 0 such that K′ = λK.2. Let θ = inf{θ ′ ∈ [0,2) : lim|x|→0|x|θ ′h(x) exists and is finite}. If γ > (n−2)24 −(2−θ)24 , then there exists c1,c2 ∈ R with c1 > 0 such thatK(x) =c1|x|β+(γ) +c2|x|β−(γ) +o(1|x|β−(γ))as x→ 0. (4.23)The ratio c2c1 is independent of the choice of K. We can therefore define themass ofΩ with respect to the operator−∆−(γ|x|2 +h(x))as mγ,h(Ω) :=c2c1.3. The mass mγ,h(Ω) satisfies the following properties:• mγ,0(Ω)< 0,• If h≤ h′ and h 6≡ h′, then mγ,h(Ω)< mγ,h′(Ω),• If Ω′ ⊂Ω, then mγ,h(Ω′)< mγ,h(Ω).Proof. The proof of (1) and (3) is similar to Proposition 2 and 4 in [106] with onlya minor change that accounts for the singularity of h. To illustrate the role of thisextra singularity we prove (2). For that, we let η ∈C∞c (Ω) be such that η(x) ≡ 1around 0. Our first objective is to write K(x) := η(x)|x|β+(γ) + f (x) for some f ∈H10 (Ω).Note that γ > (n−2)24 − (2−θ)24 ⇐⇒ β+−β−< 2−θ ⇐⇒ 2β+< n−θ . Fix θ ′ suchthat θ < θ ′ < min{2+θ2 ,2− (β+(γ)−β−(γ))}. Then lim|x|→0|x|θ ′h(x) exists and isfinite.Consider the functiong(x) =−(−∆−(γ|x|2 +h(x)))(η |x|−β+(γ)) in Ω\{0}.225Since η(x)≡ 1 around 0, we have that|g(x)| ≤∣∣∣∣ h(x)|x|β+(γ)∣∣∣∣≤C|x|−(β+(γ)+θ ′) as x→ 0. (4.24)Therefore g ∈ L 2nn+2 (Ω) if 2β+(γ)+ 2θ ′ < n+ 2, and this holds since by our as-sumption 2β+ < n− θ and 2θ ′ < 2+ θ . Since L 2nn+2 (Ω) = L 2nn−2 (Ω)′ ⊂ H10 (Ω)′,there exists f ∈ H10 (Ω) such that−∆ f −(γ|x|2 +h(x))f = g in H10 (Ω).By regularity theory, we have that f ∈C2(Ω\{0}). We now show that|x|β−(γ) f (x) has a finite limit as x→ 0. (4.25)Define K(x) = η(x)|x|β+(γ) + f (x) for all x ∈Ω\{0}, and note that K ∈C2(Ω\{0}) andis a solution to−∆K−(γ|x|2 +h(x))K = 0.Write g+(x) := max{g(x),0} and g−(x) := max{−g(x),0} so that g = g+− g−,and let f1, f2 ∈ H10 (Ω) be weak solutions to−∆ f1−(γ|x|2 +h(x))f1 = g+ and −∆ f2−(γ|x|2 +h(x))f2 = g− in H10 (Ω).(4.26)In particular, uniqueness, coercivity and the maximum principle yield f = f1− f2and f1, f2 ≥ 0. Assume that f1 6≡ 0 so that f1 > 0 in Ω \ {0}, fix α > β+(γ) andµ > 0. Define u−(x) := |x|−β−(γ)+ µ|x|−α for all x 6= 0. We then get that there226exists a small δ > 0 such that(−∆−(γ|x|2 +h(x)))u−(x)= µ(−∆− γ|x|2)|x|−α −µh(x)|x|−α −h(x)|x|−β−(γ)=−µ (α−β+(γ))(α−β−(γ))−|x|2h(x)(|x|α−β−(γ)+µ)|x|α+2< 0 for x ∈ Bδ (0)\{0},(4.27)This implies that u−(x) is a sub-solution on Bδ (0)\{0}. Let C> 0 be such that f1≥Cu− on ∂Bδ (0). Since f1 and Cu− ∈ H10 (Ω) are respectively super-solutions andsub-solutions to(−∆−(γ|x|2 +h(x)))u(x) = 0, it follows from the comparisonprinciple (via coercivity) that f1 >Cu−>C|x|−β−(γ) on Bδ (0)\{0}. It then followsfrom (4.24) thatg+(x)≤ |g(x)| ≤C|x|−(β+(γ)+θ ′) ≤C1|x|(2−θ ′)−(β+(γ)−β−(γ)) f1|x|2 .Then rewriting (4.26) as−∆ f1−(γ|x|2 +h(x)+g+f1)f1 = 0yields−∆ f1−γ+O(|x|(2−θ ′)−(β+(γ)−β−(γ)))|x|2 f1 = 0.With our choice of θ ′ we can then conclude by the optimal regularity result in [106,Theorem 8] that |x|β−(γ) f1 has a finite limit as x→ 0. Similarly one also obtains that|x|β−(γ) f2 has a finite limit as x→ 0, and therefore (4.25) is verified.It follows that there exists c2 ∈ R such thatK(x) =1|x|β+(γ) +c2|x|β−(γ) +o(1|x|β−(γ))as x→ 0,227which proves the existence of a solution K to the problem with the relevant asymp-totic behavior. The uniqueness result yields the conclusion.We now proceed with the proof of the existence results, following again [106].We shall use the following standard sufficient condition for attainability.Lemma 4.4.2. Under the assumptions of Theorem 4.1.5, ifµγ,h(Ω) := infu∈H10 (Ω)\{0}∫Ω(|∇u|2−(γ|x|2 +h(x))u2)dx( ∫Ωb(x)|u|2∗(s)|x|s dx)2/2∗(s) < µγ,0(Rn)b(0)2/2∗(s) ,then the infimum µγ,s(Ω) is achieved and equation (4.19) has a solution.Proof of Theorem 4.1.5: We will construct a minimizing sequence uε in H10 (Ω)\{0} for the functional JΩγ,h in such a way that µγ,h(Ω) < b(0)−2/2∗(s)µγ,0(Rn). Asmentioned above, when γ ≥ 0 the infimum µγ,0(Rn) is achieved, up to a constant,by the functionU(x) :=1(|x| (2−s)β−(γ)n−2 + |x| (2−s)β+(γ)n−2) n−22−sfor x ∈ Rn \{0}.In particular, there exists χ > 0 such that−∆U− γ|x|2U = χU2∗(s)−1|x|s in Rn \{0}. (4.28)Define a scaled version of U byUε(x) := ε−n−22 U( xε)=(ε2−sn−2 ·β+(γ)−β−(γ)2ε2−sn−2 ·(β+(γ)−β−(γ))|x| (2−s)β−(γ)n−2 + |x| (2−s)β+(γ)n−2) n−22−s(4.29)for x ∈ Rn \{0}. β±(γ) are defined in (4.16). In the sequel, we write β+ := β+(γ)and β− := β−(γ). Consider a cut-off function η ∈C∞c (Ω) such that η(x) ≡ 1 in aneighborhood of 0 contained in Ω.228Case 1: Test-functions for the case when γ ≤ (n−2)24− (2−θ)24.For ε > 0, we consider the test functions uε ∈ D1,2(Ω) defined by uε(x) :=η(x)Uε(x) for x ∈Ω\{0}. To estimate JΩγ,h(uε), we use the bounds on Uε to obtain∫Ωb(x)u2∗(s)ε|x|s dx =∫Bδ (0)b(x)U2∗(s)ε|x|s dx+∫Ω\Bδ (0)b(x)u2∗(s)ε|x|s dx=∫Bε−1δ (0)b(εx)U2∗(s)|x|s dx+∫Bε−1δ (0)b(εx)η(εx)2∗(s)U2∗(s)|x|s dx= b(0)∫RnU2∗(s)|x|s dx+O(ε2∗(s)2 (β+−β−)).Similarly, one also has∫Ω(|∇uε |2− γ|x|2 u2ε)dx=∫Bδ (0)(|∇Uε |2− γ|x|2U2ε)dx+∫Ω\Bδ (0)(|∇uε |2− γ|x|2 u2ε)dx=∫Bε−1δ (0)(|∇U |2− γ|x|2U2)dx+O(εβ+−β−)=∫Rn(|∇U |2− γ|x|2U2)dx+O(εβ+−β−)= χ∫RnU2∗(s)|x|s dx+O(εβ+−β−).Estimating the lower order terms as ε → 0 gives∫Ωh˜(x)u2ε dx =ε2−θ[C2∫RnU2|x|θ dx+o(1)]if β+−β− > 2−θ ,ε2−θ log(1ε)[C2ωn−1+o(1)] if β+−β− = 2−θ ,O(εβ+−β−)if β+−β− < 2−θ .229And−C1∫Ωlog |x||x|θ u2ε dx =C1ε2−θ log( 1ε)[ ∫RnU2|x|θ dx+o(1)]if β+−β− > 2−θ ,C1ε2−θ(log(1ε))2 [ ωn−12+o(1)]if β+−β− = 2−θ ,O(εβ+−β−)if β+−β− < 2−θ .Note that β+−β− ≥ 2−θ if and only if γ ≤ (n−2)24 − (2−θ)24 . Therefore,∫Ωh(x)u2ε dx =ε2−θ∫RnU2|x|θ dx[C1 log(1ε)(1+o(1))+C2+o(1)]if γ < (n−2)24 − (2−θ)24 ,ε2−θ log(1ε)ωn−12[C1 log(1ε)(1+o(1))+2C2+o(1)]if γ = (n−2)24 − (2−θ)24 .230Combining the above estimates, we obtain as ε → 0,JΩγ,h(uε)=∫Ω(|∇uε |2− γ u2ε|x|2 −h(x)u2ε)dx( ∫Ω b(x)|uε |2∗(s)|x|s dx)2/2∗(s)=µγ,0(Rn)b(0)2/2∗(s)−(∫RnU2|x|θ dx)ε2−θ(b(0)∫RnU2∗(s)|x|s dx)2/2∗(s) [C1 log( 1ε )(1+o(1))+C2+o(1)]if γ < (n−2)24 − (2−θ)24 ,ωn−1ε2−θ log( 1ε )2(b(0)∫RnU2∗(s)|x|s dx)2/2∗(s) [C1 log( 1ε )(1+o(1))+2C2+o(1)]if γ = (n−2)24 − (2−θ)24 ,as long as β+− β− ≥ 2− θ . Thus, for ε sufficiently small, the assumption thateither C1 > 0 or C1 = 0, C2 > 0 guarantees thatµγ,h(Ω)≤ JΩγ,h(uε)<µγ,0(Rn)b(0)2/2∗(s).It then follows from Lemma 4.4.2 that µγ,h(Ω) is attained.Case 2: Test-functions for the case when(n−2)24− (2−θ)24< γ <(n−2)24.Here h(x) and θ given by (4.11) satisfy the hypothesis of Proposition (4.4.1). Sinceγ > (n−2)24 − (2−θ)24 , it follows from (4.23) that there exists β ∈ D1,2(Ω) such thatβ (x)'x→0 mγ,h(Ω)|x|β− . (4.30)The function K(x) := η(x)|x|β+ +β (x) for x ∈Ω\{0} satisfies the equation:−∆K−(γ|x|2 +h(x))K = 0 in Ω\{0}K > 0 in Ω\{0}K = 0 on ∂Ω.(4.31)231Define the test functionsuε(x) := η(x)Uε + εβ+−β−2 β (x) for x ∈Ω\{0}The functions uε ∈ D1,2(Ω) for all ε > 0. We estimate JΩγ,h(uε).Step 1: Estimates for∫Ω(|∇uε |2−(γ|x|2 +h(x))u2ε)dx.Take δ > 0 small enough such that η(x) = 1 in Bδ (0) ⊂ Ω. We decompose theintegral as∫Ω(|∇uε |2−(γ|x|2 +h(x))u2ε)dx=∫Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx+∫Ω\Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx.By standard elliptic estimates, it follows that limε→0 uεεβ+−β−2= K in C2loc(Ω\{0}).Hencelimε→0∫Ω\Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dxεβ+−β−=∫Ω\Bδ (0)(|∇K|2−(γ|x|2 +h(x))K2)dx=∫Ω\Bδ (0)(−∆K−(γ|x|2 +h(x))K)K dx+∫∂ (Ω\Bδ (0))K∂νK dσ=∫∂ (Ω\Bδ (0))K∂νK dσ =−∫∂Bδ (0)K∂νK dσ .Since β++β− = n−2, using elliptic estimates, and the definition of K gives usK∂νK =− β+|x|1+2β+ − (n−2)mγ,h(Ω)|x|n−1 +o(1|x|n−1)as x→ 0.232Therefore,∫Ω\Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx= εβ+−β−ωn−1(β+δ β+−β−+(n−2)mγ,h(Ω)+oδ (1))Now, we estimate the term∫Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx.First, uε(x) = Uε(x) + εβ+−β−2 β (x) for x ∈ Bδ (0), therefore after integration byparts, we obtain∫Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx=∫Bδ (0)(|∇Uε |2−(γ|x|2 +h(x))U2ε)dx+2εβ+−β−2∫Bδ (0)(∇Uε ·∇β −(γ|x|2 +h(x))Uεβ)dx+ εβ+−β−∫Bδ (0)(|∇β |2−(γ|x|2 +h(x))β 2)dx=∫Bδ (0)(−∆Uε − γ|x|2Uε)Uε dx+∫∂Bδ (0)Uε∂νUε dσ−∫Bδ (0)h(x)U2ε dx+2εβ+−β−2∫Bδ (0)(−∆Uε dx− γ|x|2Uε)β dx−2ε β+−β−2∫Bδ (0)h(x)Uεβ dx+2εβ+−β−2∫∂Bδ (0)β∂νUε dσ+ εβ+−β−∫Bδ (0)(|∇β |2−(γ|x|2 +h(x))β 2)dx.We now estimate each of the above terms. First, using equation (4.20) and theexpression for Uε defined as in (4.29), we obtain∫Bδ (0)(−∆Uε − γ|x|2Uε)Uε dx = χ∫Bδ (0)U2∗(s)ε|x|s dx= χ∫RnU2∗(s)|x|s dx+O(ε2∗(s)2 (β+−β−)),233and ∫∂Bδ (0)Uε∂νUε dσ =−β+ωn−1 εβ+−β−δ β+−β−+oδ(εβ+−β−)as ε → 0.Note thatβ+−β− < 2−θ ⇐⇒ γ > (n−2)24− (2−θ)24=⇒ 2β++θ < n.Therefore,∫Bδ (0)h(x)U2ε dx = O(εβ+−β−∫Bδ (0)1|x|2β++θ dx)= oδ(εβ+−β−)as ε → 0.Again from equation (4.20) and the expression for U and β , we get that∫Bδ (0)(−∆Uε dx− γ|x|2Uε)β dx= εβ++β−2∫Bε−1δ (0)(−∆U dx− γ|x|2U)β (εx) dx= mγ,h(Ω)εβ+−β−2∫Bε−1δ (0)(−∆U dx− γ|x|2U)|x|−β− dx+oδ(εβ+−β−2)= mγ,h(Ω)εβ+−β−2∫Bε−1δ (0)(−∆|x|−β− dx− γ|x|2 |x|−β−)U dx−mγ,h(Ω)εβ+−β−2∫∂Bε−1δ (0)∂νU|x|β− dσ +oδ(εβ+−β−2)= β+mγ,h(Ω)ωn−1εβ+−β−2 +oδ(εβ+−β−2).Similarly,∫∂Bδ (0)β∂νUε dσ =−β+mγ,h(Ω)ωn−1εβ+−β−2 +oδ(εβ+−β−2).234Since β++β−+θ = n− (2−θ)< n, we have∫Bδ (0)h(x)Uεβ dx = O(εβ+−β−2∫Bδ (0)1|x|β++β−+θ dx)= oδ(εβ+−β−2).And, finallyεβ+−β−∫Bδ (0)(|∇β |2−(γ|x|2 +h(x))β 2)dx = oδ (εβ+−β−).Combining all the estimates, we get∫Bδ (0)(|∇uε |2−(γ|x|2 +h(x))u2ε)dx= χ∫RnU2∗(s)|x|s dx−β+ωn−1εβ+−β−δ β+−β−+oδ (εβ+−β−).So,∫Ω(|∇uε |2−(γ|x|2 +h(x))u2ε)dx= χ∫RnU2∗(s)|x|s dx+ωn−1(n−2)mγ,h(Ω)εβ+−β−+oδ (εβ+−β−).Step 2: Estimating∫Ωb(x)u2∗(s)ε|x|s dx.235One has for δ > 0 small∫Ωb(x)u2∗(s)ε|x|s dx=∫Bδ (0)b(x)u2∗(s)ε|x|s dx+∫Ω\Bδ (0)b(x)u2∗(s)ε|x|s dx=∫Bδ (0)b(x)(Uε(x)+ εβ+−β−2 β (x))2∗(s)|x|s dx+o(εβ+−β−)=∫Bδ (0)b(x)U2∗(s)ε|x|s dx+ εβ+−β−2 2∗(s)∫Bδ (0)b(x)U2∗(s)−1ε|x|s β dx+o(εβ+−β−)=∫Bδ (0)b(x)U2∗(s)ε|x|s dx+ εβ+−β−22∗(s)χ∫Bδ (0)b(x)(−∆Uε dx− γ|x|2Uε)β dx+o(εβ+−β−)= b(0)∫RnU2∗(s)|x|s dx+2∗(s)χb(0)β+mγ,λ ,a(Ω)ωn−1εβ+−β−+o(εβ+−β−).So, we obtainJΩγ,λ ,a(uε) (4.32)=∫Ω(|∇uε |2− γ u2ε|x|2 −h(x)u2ε)dx( ∫Ωb(x)|uε |2∗(s)|x|s dx)2/2∗(s)=µγ,0(Rn)b(0)2/2∗(s)−mγ,h(Ω) ωn−1(β+−β−)(b(0)∫RnU2∗(s)|x|s dx)2/2∗(s) εβ+−β−+o(εβ+−β−).(4.33)Therefore, if mγ,h(Ω)> 0, we get for ε sufficiently smallµγ,h(Ω)≤ JΩγ,h(uε)<µγ,0(Rn)b(0)2/2∗(s).236Then, from Lemma 4.4.2 it follows that µγ,h(Ω) is attained. Remark 4.4.3. Assume for simplicity that h(x) = λ |x|−θ where 0≤ θ < 2. There isa threshold λ ∗(Ω)≥ 0 beyond which the infimum µγ,λ (Ω) is achieved, and belowwhich, it is not. In fact,λ ∗(Ω) := sup{λ : µγ,λ (Ω) = µγ,0(Rn)}.Performing a blow-up analysis like in [106] one can obtain the following sharpresults:• In high dimensions, that is for γ ≤ (n−2)24 − (2−θ)24 one has λ∗(Ω) = 0 andthe infimum µγ,λ (Ω) is achieved if and only if λ > λ ∗(Ω).• In low dimensions, that is for (n−2)24 − (2−θ)24 < γ , one has λ∗(Ω) > 0 andµγ,λ (Ω) is not achieved for λ < λ ∗(Ω) and µγ,λ (Ω) is achieved for λ >λ ∗(Ω). Moreover under the assumption µγ,λ ∗(Ω) is not achieved, we havethat mγ,λ ∗(Ω) = 0, and λ ∗(Ω) = sup{λ : mγ,λ (Ω)≤ 0}.4.5 Existence results for compact submanifolds of BnConsider the following Dirichlet boundary value problem in hyperbolic space. LetΩ b Bn (n ≥ 3) be a bounded smooth domain such that 0 ∈ Ω. We consider theDirichlet boundary value problem:−∆Bnu− γV2u−λu =V2∗(s)u2∗(s)−1 in Ωu≥ 0 in Ωu = 0 on ∂Ω,(4.34)where λ ∈ R, 0 < s < 2 and γ < γH := (n−2)24 .We shall use the conformal transformation gBn = ϕ4n−2 gEucl, whereϕ =(21−r2) n−22to map the problem into Rn. We start by considering the237general equation :−∆Bnu− γV2u−λu = F(x,u) in Ωb Bn, (4.35)where F(x,u) is a Carathe´odory function such that|F(x,u)| ≤C|u|(1+|u|2∗(s)−2rs)for all x ∈Ω.If u satisfies (4.35), then v := ϕu satisfies the equation:−∆v− γ(21− r2)2V2v−[λ − n(n−2)4](21− r2)2v = ϕn+2n−2 f(x,vϕ)in Ω.On the other hand, we have the following expansion for(21−r2)2V2 :(21− r2)2V2(x) =1(n−2)2(f (r)G(r))2where f (r) and G(r) are given by (4.1). We then obtain that(21− r2)2V2(x) =1r2 +4r +8+g3(r) when n = 3,1r2 +8log1r −4+g4(r) when n = 4,1r2 +4(n−2)n−4 + rgn(r) when n≥ 5.(4.36)where for all n≥ 3, gn(0) = 0 and gn is C0([0,δ ]) for δ < 1.This implies that v := ϕu is a solution to−∆v− γr2v−[γa(x)+(λ − n(n−2)4)(21− r2)2]v = ϕn+2n−2 f(x,vϕ).where a(x) is defined in (4.8). We can therefore state the following lemma:238Lemma 4.5.1. A non-negative function u ∈ H10 (Ω) solves (4.34) if and only ifv := ϕu ∈ H10 (Ω) satisfies−∆v−(γ|x|2 +hγ,λ (x))v = b(x) v2∗(s)−1|x|s in Ωv≥ 0 in Ωv = 0 on ∂Ω,(4.37)wherehγ,λ (x) = γa(x)+4λ −n(n−2)(1−|x|2)2 ,a(x) is defined in (4.8), and b(x) is a positive function in C0(Ω) with b(0) =(n−2) n−sn−222−s. Moreover, the hyperbolic operator LBnγ :=−∆Bn − γV2−λ is coerciveif and only if the corresponding Euclidean operator LRnγ,h :=−∆−(γ|x|2 +hγ,λ (x))is coercive.Proof. Note that one has in particularhγ,λ (x) = hγ,λ (r) =4γr +8γ+4λ−3(1−r2)2 + γg3(r) when n = 3,[8γ log 1r −4γ+4λ −8]+γg4(r)+(4λ −8) r2(2−r2)(1−r2)2 when n = 4,4(n−2)n−4[n−4n−2λ + γ− n(n−4)4]+γrgn(r)+(4λ −n(n−2)) r2(2−r2)(1−r2)2 when n≥ 5,(4.38)with gn(0) = 0 and gn is C0([0,δ ]) for δ < 1, for all n≥ 3.Let F(x,u) = V2∗(s)u2∗(s)−1 in (4.35). The above remarks show that v := ϕu isa solution to (4.37).For the second part, we first note that the following identities hold:∫Ω(|∇Bnu|2− n(n−2)4 u2)dvgBn =∫Ω|∇v|2 dx239and ∫Ωu2dvgBn =∫Ωv2(21− r2)2dx.If the operator LBnγ is coercive, then for any u ∈ C∞(Ω), we have 〈LBnγ u,u〉 ≥C‖u‖2H10 (Ω), which means∫Ω(|∇Bnu|2− γV2u2)dvgBn ≥C∫Ω(|∇Bnu|2+u2)dvgBn .The latter then holds if and only if〈LRnγ,φu,u〉=∫Ω(|∇v|2−(21− r2)2(γV2− n(n−2)4)v2)dx≥C∫Ω(|∇v|2+(21− r2)2(n(n−2)4+1)v2)dx≥C′∫Ω(|∇v|2+ v2)dx≥ c‖u‖2H10 (Ω),where v = ϕu is in C∞(Ω). This completes the proof.At this point, the proof of Theorems 4.1.2 and 4.1.3 follows verbatim as in theEuclidean case.One can then use the results obtained in the last section to prove Theorem 4.1.4stated in the introduction for the hyperbolic space. Indeed, it suffices to considerequation (4.37), where b is a positive function in C1(Ω) satisfying (4.7) and hγ,λ isgiven by (4.38).If n ≥ 5, then lim|x|→0hγ,λ (x) =4(n−2)n−4[n−4n−2λ + γ− n(n−4)4], which is positiveprovidedλ >n−2n−4(n(n−4)4− γ).Moreover, since in this case θ = 0, the first alternative in Theorem 4.1.5 holdswhen γ ≤ (n−2)24 − 1 = n(n−4)4 . For (n−2)24 − 1 < γ < (n−2)24 , the existence of theextremal is guaranteed by the positivity of the hyperbolic mass mHγ,λ (Ω) associatedto the operator LBnγ , which is a positive multiple of the mass of the correspondingEuclidean operator.240When n= 4, we can use the first option in Theorem 4.1.5 using the logarithmicperturbation iflim|x|→0(log1|x|)−1hγ,λ (x) = 8γ > 0and, since θ = 0,γ ≤ (4−2)24−1 = 0.This is impossible. In the absence of the dominating term with log 1|x| , i.e. whenγ = 0, we get existence of the extremal if λ > 4(4−2)4 = 2. Otherwise, we requirethe positivity of the hyperbolic mass mHγ,λ .Similarly, if n = 3, the threshold for γ with the singular perturbation 1|x| (i.e.θ = 1) is γ ≤ (3−2)24 − 14 = 0. In order to use the first option in Theorem 4.1.5, wehave to resort to the next term 4λ − 3, which is positive when λ > 34 , in the caseγ = 0. When γ > 0 or λ ≤ 34 , one needs that mHγ,λ > 0.241Bibliography[1] N. Abatangelo, E. Valdinoci. A notion of nonlocal curvature. 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