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Applications of entire function theory to an imbedding theorem for differentiable functions of several… Foster, David Larry 1973

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APPLICATIONS OF ENTIRE FUNCTION THEORY TO AN IMBEDDING THEOREM FOR DIFFERENTIABLE FUNCTIONS OF SEVERAL REAL VARIABLES by DAVID LARRY FOSTER B.S., Iowa State U n i v e r s i t y , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Mathematics We accept t h i s t h e s i s as conforming t o the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1973 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date \JU g i i ABSTRACT The subject o f t h i s t h e s i s i s the f r a c t i o n a l order Sobolev space, Hp, as considered by N i k o l ' s k i i ; the go a l i s t o demonstrate an imbedding theorem f o r Hp* analogous t o the c l a s s i c a l imbedding theorem f o r W*^  which was f i r s t shown by Sobolev. r The p r o p e r t i e s e s t a b l i s h e d here f o r spaces Hp defined over a l l of R n, i n c l u d i n g completeness and imbedding theorems, are demonstrated by a technique i n v o l v i n g the approximation o f fu n c t i o n s i n those spaces by e n t i r e f u n c t i o n s o f the exp o n e n t i a l type. P r o p e r t i e s o f such e n t i r e f u n c t i o n s , which are of i n t e r e s t i n t h e i r e own r i g h t , are developed i n a separate chapter. An extension theorem f o r d i f f e r e n t i a b l e f u n c t i o n s defined over an ©pen subset o f R n i s a l s o proved. i i i TABLE OF CONTENTS Page CHAPTER ONE INTRODUCTION 1 CHAPTER TWO DEFINITIONS 3 CHAPTER THREE AN EXTENSION "THEOREM 8 CHAPTER FOUR ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE . . 19 CHAPTER FIVE APPROXIMATION OF CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE . . 30 CHAPTER SIX THE NORMED SPACE U3 P -•• CHAPTER SEVEN AN IMBEDDING THEOREM'FOR THE SPACE H£ . . . 53 CHAPTER EIGHT THE CLASSES HJJ(G n) , w £ ( O n ) , B^( G Q) 62 BIBLIOGRAPHY 66 ACKNOWLEDGEMENTS I wish to express my grateful appreciation to Dr. R. A. Adams, my thesis supervisor, for introducing me to Soholev Spaces, and for his continual assistance, guidance, and encouragement during the writing of this thesis. I also wish to thank Dr. J. Foumier for his help and assistance in reading this work. The finanacial assistance of the University of British Columbia and the National Research Council i s also acknowledged. CHAPTER ONE 1 INTRODUCTION The Sobolev space ^ ( G n ) of f u n c t i o n s f , i n t e g r a b l e on a domain G n of R11 and having there a l l g e n e r a l i z e d p a r t i a l d e r i v a t i v e s of order m i n -t e g r a b l e t o the power p >_ 1, i s w e l l known, and a great many r e s u l t s con-c e r n i n g Sobolev spaces are being c o n t i n u a l l y formulated. We no t e , however, the incompleteness o f t h i s concept by the r e s t r i c t i o n t h a t the parameter m be an i n t e g e r . In t h i s work we examine a Sobolev space o f f r a c t i o n a l order and i t s p r o p e r t i e s . We give a survey of some of the many papers by S. N. N i k o l ' s k i i on the space ^ ( G ^ ) where r i s allowed any r e a l value. In p a r t i c u l a r we i n v e s t i g a t e a l l p o s s i b l e imbeddings f o r the space H^, and examine the r e -l a t i o n s h i p between t h i s and the t r a d i t i o n a l Sobolev space i n the case r = m an i n t e g e r . The technique o f t h i s i n v e s t i g a t i o n i s b a s i c a l l y the approximation of continuous f u n c t i o n s by e n t i r e f u n c t i o n s o f the e x p o n e n t i a l type. We f i r s t give the c o n d i t i o n s f o r the extension o f a f u n c t i o n f e H p(G n) from G t o the e n t i r e space R n, and subsequently t r e a t only the c l a s s H r(B n). We next d i s p l a y some relevant p r o p e r t i e s of e n t i r e f u n c t i o n s of exponential t y p e , and then give necessary and s u f f i c i e n t c o n d i t i o n s f o r the approx-imation of a f u n c t i o n f belonging t o HptR 1 3) by such e n t i r e f u n c t i o n s ; the demonstration of a l l subsequent r e s u l t s i s based on t h i s approximation. Completeness i s given f o r H^R 1 1) w i t h a us u a l norm, and the p r i n c i p a l r e s u l t o f t h i s work, the imbedding theorem, i s proved i n Chapter Seven. 2 Finally, brief mention is made of the spaces w £ and Bp where the parameter r i s again allowed any positive real value. We note that the results given here are not precisely of the same form as dealt with by Nikol'skii. In particular the definition of the space Hp* presented here i s not the usual one. We have made some changes both in definitions end in proofs in order to f u l l y elab-orate on the beauty and precision of the approximation of continuous functions by entire functions. 3 CHAPTER TWO DEFINITIONS We make the following notational conventions: I f M i s a normed linear space and f e M, then the norm of f in M i s denoted by ||f : M||. Gn denotes a open domain in Rn- n dimensional Euclidean space; Gn ^ i s the set of points in Gn at distance greater than n from the boundary of G ; G* _ i s the set of points in G at distance less than l/n from the n' n,n * n,n 3x i origin. (k) If f = f(x 1,** ,,x n) is a function defined on G n, then f£ ' and Aj.(f,h) = f ( x l t « * ' t 3 t n ) . A^(f,h) = ^ ( f . h ) - fU-L/'-.Xi + h,***,xn) - fCx^y.Xn), A* (f,h) « A x.(A^' 1(f,h)) for k = 2, 3, * i A i x^ Note that A^ (f,h) i s defined on G provided that k|h| < n. x«£ n ^  n Now to define the derivative, we say that the function f = t(x^t'" txn) has a par t i a l derivative on the region Gn i f f may be altered on a a P - l f 3xJ n set of measure zero in such a manner that the partial derivative ax?- 1 exists and i s absolutely continuous with respect to x^ on any closed seg-ment parallel to the axis lying in G n (i.e. any segment in G on a line with fixed coordinates (x^,* • • »Xi_i»x i + 1,* • • .Xjj) ). The partial derivative Is thus defined uniquely up to a set of measure zero. k The mixed p a r t i a l , p ff" pn < p l + p2 + — + P n - P> i s d e f i n e d by i n d u c t i o n . Indeed, i f the p a r t i a l d e r i v a t i v e aP" p«f Y " T - F ; — , p s - i 3 x ; i — a x g V e x i s t s almost everywhere i n G n and i f i t i s p o s s i b l e t o a l t e r V on a set of measure zero so th a t f o r a l l x^,' • • *xs_i»xs-KL»" * * » x n * n a s 8 1 1 a b s o l u t e l y continuous p a r t i a l d e r i v a t i v e of order p s-l w i t h respect t o x g (on any c l o s e d segment i n G n p a r a l l e l t o the x s a x i s ) , the p a r t i a l d e r i v a t i v e e x i s t s i n the sense described above, and we denote i t as 9Py _ aPs ^ aP-Psf j 3 x l P l " * 3 x s P s " 8x7^  3x1P1.--3xsy-1 Note however, that t h i s d e f i n i t i o n does not correspond t o t h a t o f the ge n e r a l i z e d d e r i v a t i v e u s u a l l y used i n d e f i n i n g Sobolev spaces ( c f . Sobolev [ l U ] ) . To view t h i s work as a g e n e r a l i z a t i o n o f the work o f Sobolev, we 3 p f show i n Chapter Seven th a t the nonmixed p a r t i a l d e r i v a t i v e s- defined 3x. I here c o i n c i d e s w i t h the g e n e r a l i z e d d e r i v a t i v e , provided t h a t both the fu n c t i o n and i t s d e r i v a t i v e are i n t e g r a b l e on G n. While t h i s does not h o l d f o r a l l mixed p a r t i a l s , i f the f u n c t i o n f i s i n t e g r a b l e on G Q, and i f the d e r i v a t i v e s 3 p l f f 3 P l ^ p 2 f t ...t 3 p f 9x x 1 zx^h^2 a x ^ i - ' - a x j ! 0 1 1 e x i s t and are i n t e g r a b l e on G n, then these d e r i v a t i v e s are the corresponding g e n e r a l i z e d d e r i v a t i v e s . Conversely, i f the f u n c t i o n f i s summable on G and has a l l the g e n e r a l i z e d d e r i v a t i v e s , they are the d e r i v a t i v e s as de-n f i n e d here. 5 For the remainder o f t h i s work, unless otherwise s p e c i f i e d , we l e t p >_ 1, r > 0 and w r i t e r = p + a where p i s an i n t e g e r , and 0 < a <. 1. We denote by r a v e c t o r ( r 1 , , " , r n ; w i t h each = + p^ as above. We say t h a t the f u n c t i o n f = f t x ^ , * * * , ^ ) belongs t o the c l a s s H^^ X l(G n,M) i f f i s de f i n e d on G n, i n t e g r a b l e together w i t h i t s p a r t i a l d e r i v a t i v e s 3 k f (k = 1, 2 , •••, p) i n the p-th power on G , and i f . f u r t h e r -more, f o r every n > 0 such t h a t G 4 0 we have » n ,n where h i s an a r b i t r a r y r e a l number s a t i s f y i n g 2|h| < n. We define the c l a s s „ (Gn,M) f o r i = 2, 3, • " , n analogously. P» xi 1 1 I f f belongs t o a l l the c l a s s e s H p n X i » M i ^ f o r * = ^» 2 » **** n * then l e t t i n g r = ( r ^ r g , * *' , r n ) and M = (M^Mg,'• * , we say t h a t f belongs t o the c l a s s Hp*(Gn,M). I f G n = R11, then we note t h a t G n ^ = R and w r i t e Hp*(M") f o r Hp(R n,M). A l s o , we w r i t e f e Hp*(Gn) i f f e Hj(G n,M) f o r some M, and s i m i l a r l y f e Hp i f f e HJ(M') f o r some M. j» We note t h a t the c o n d i t i o n ( l ) i n the d e f i n i t i o n o f H_ V j can be P * x i , , ( P ) , s i m p l i f i e d i n the case a < 1 by r e p l a c i n g the second d i f f e r e n c e A ^ v f ^ » n' by the f i r s t d i f f e r e n c e , A„.(fY. ,h). This s i m p l i f i c a t i o n i s made i n c e r t a i n circumstances t o conform w i t h a more general usage* f o r ease of c a l c u l a t i o n under the i n t e g r a l , and i n the d e f i n i t i o n o f the norm of the space Hp^Gjj). The equivalence of the two d e f i n i t i o n s w i l l be discussed i n Chapter F i v e . In the case G Q = R n, however, the equivalence o f the two d e f i n i t i o n s ( f o r a < l ) may a c t u a l l y be proved d i r e c t l y i n the f o l l o w i n g manner: From our d e f i n i t i o n f o r a f u n c t i o n o f one v a r i a b l e , 6 A 2 ( f , h ) • A(A(f,h)) • f ( x + 2h) - 2 f ( x +h) • fix) whence we see t h a t n-1 , , A(f,h) - M(tt2%) - - | I 4 A 2 ( f , 2 J h ) (2) which may be v e r i f i e d by expanding the r i g h t hand s i d e . LEMMA 1. I f f e Lp(R) and 0 < o < 1, then the f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t : i ) | | A ( f , h ) : Lp(R)|| < c|h| a , i i ) | | A 2 ( f , h ) : Lp(R)|| < c»|h|°. PROOF. C l e a r l y ( i ) i m p l i e s ( i i ) . To show the converse, assume ( i i ) h o l d s . Since f e I^(R) 11-1 A(f,2 nh>: LpOOM 1 £ C (. / R | f ( x ) | P d x ) 1 / p + ( J R | f ( x • 2 n h » P d x ) 1 / P ] 8 8 ^ E = T I l f : VR )" 0 a s n "*0 * Then from (2) we have | | A ( f , h ) : L p ( R ) | | < | | i T^gi A 2 ( f , 2 J h ) : Lp(H)|| < £ I 1 | | A 2 ( f , 2 J h ) : L _ ( R ) | | * j=0 2 J p l l j ^ c ^ N - c M h l ^ ^ - ^ ) ( 3 ) 2 a ( 2 1 - a i l ) h|° - c | h | \ which proves t h a t ( i i ) i m p l i e s ( i ) . Note t h a t ( i i ) may be replaced by i i i ) [ / | f ( x + h) - 2 f ( x ) + f ( x - h)| Pdx ] 1 / P < c'|h| S — by t r a n s l a t i o n i n the i n t e g r a l . 7 In the case a = i , however, i t i s necessary t o r e t a i n the second d i f f e r e n c e , A x j ^ ^ x i * * 1 ) * n ^ m T h i s me>-y he seen from l i n e (3) above, or from l i n e s (20) and (21) o f Chapter F i v e . In the a c t u a l c a l c u l a t i o n o f t h i s work, we use the f i r s t d i f f e r e n c e A ( f ^ , h ) (when o < l ) only i n Chapter S i x which deals w i t h the norm x i x i of the Space. For the s i m p l i c i t y o f avoiding the c o n s i d e r a t i o n o f separate 9 (p) . cases f o r a <1 and a* 1, we use A^ ( f ,h) elsewhere. x i x i 8 CHAPTER THREE AN EXTENSION THEOREM The r e s u l t o f t h i s chapter i s a f a m i l i a r one which provides an ex-r*. * n t e n s i o n of f e H-,(G -M) from G_ t o K p r e s e r v i n g what i s e s s e n t i a l l y the norm of f• Thus, i n future chapters we w i l l d e a l only w i t h the c l a s s Hp(M) o f f u n c t i o n s d e f i n e d on a l l o f K . A l s o note t h a t we place no smooth-ness c o n d i t i o n s on the boundary o f G n, but w i l l always d e a l w i t h P n ,n o r On,n w n i c n e l i m i n a t e any re l e v a n t i r r e g u l a r i t i e s on the boundary. We begin w i t h an i n t e r e s t i n g and e s s e n t i a l lemma whose proof w e l l i l l u s t r a t e s the s i g n i f i c a n c e o f the Holder c o n d i t i o n i n the d e f i n i t i o n of the c l a s s Hp". LEMMA 1. Let f e Hp^ x^(G n,M). For any constant n >_ 0, there e x i s t s con-s t a n t s c - j ^ n and eg n depending only on n so th a t the i n e q u a l i t y M^' Wn'Hi c i , J | f ! LP(0n>M * * 2 > n M <il 1 holds f o r a l l k = 1, 2, p. PROOF. For almost a l l ( x 2 , " * , x n ) allowed i n G n the f u n c t i o n f may be expanded i n the form f(x 1+h,X2,'" ,x n) = f(x 1,««»,x n) + hJL tUlt"' »x n) + ••• + 3x x (2) + ^ . - 4 f ( X l , - ' - , x n ) + R(h) p! 3x* where R ( h ) = _ 1 _ / h ( h . t ) ^ 1 [ a P f ^ n ^ o t — ^ n ) . (p-l)l 0 3xf (3) _ f f(xi,...txn) 3 d t ^  3 X j 9 This expansion i s a form of Taylor's lformula vhere the remainder (3) can be verified in the same manner as the usual remainder. The equation (2) holds for almost a l l (xj_,* *' j X ^ ) in G n > r i/2 m A f o r a 1 1 h i i / t n M * n/ 2 . Thus we may replace h by -h in (2) to get f(x 1-h tx 2,---,x n) = f - h - i l + ••• + (-i)piiii!£ + 3x^ p ! 3x? + Isill [ / (h - t ) P * 1 ( 3 pfCxi-t tX2,...,x nQ 1 _ ( p - 1 ) ! 0 3x1 3 p f ( x i , . . . , x n ) - — — — — — j dt. 3xJ F i r s t suppose that p i s even. Then adding equations (2) and (h) we have f U ^ h . X g , * " .Xj,) + f U-L-h.Xoy.Xn) = = 2(f + ••• + — ) + R#Ch) 2! 3x 2 p ! 3xJ where R # ( h ) = J h ( h - t ) p - 1 [ ^ I ^ S " - ^ ) ( p - 1 ) ! 0 "3x£ 30f(xi,-«« ,xn) 3 p f ( x i - t , x 2 ^ - - s x n ) , - 2 j — ^ - + - ] dt. 3xp 3x P (U) (5) (6) In equation (5) transfer the remainder R #(h) to the l e f t hand side of the equation, and pick distinct real numbers h Q , n p ^ 2 satisfying |h^| < ,n/2. Then upon substitution of the numbers into the new equation.^ we obtain a system of p/2 + 1 linear equations in the variables f, 3 2f 3P-P —-jjr, *"'* This system has a non-singular determinent, and we solve 3x 1 Sx-^  the system as 10 - 1 32k, 0/2 3 x f E Y i k [ f ( V h i » X 2 » ' " » X n ) " i=0 1 » K ( k = 0, 1, p/2), where the y^ ^'s are numbers depending on the h^'s. Consequently 3 2 k f P/2 l i r a : y*n.n/2>N * < J I .1. [f(x 1thi,x2,---,xn) -9 x l Gn,n/2 1 = 0 - f(x-L-b^ ,x 2^• • • ,x n) - R^h^) ]| Pdx J} < P f IY± k l M J I K x i + h i ^ . - " . ^ ) ! 5 * ) 1 / p • + ( I i f C x ^ h i ^ , — , x n ) | p d x ) 1 / p + ( / IR.^)!^)1^ > Gn, n/2 Gn,n/2 1 * ^ 2 T |YLIK| tlf v S ( G n , n / 2 ) N * p/2 , h A , 3 p f ( x 1 t t , x 2 f - " , x n ) i-o ^ (P-D! 0 G n f T l / 2 3xJ 3 p f ( x 1 , ' " , x n ) ^ f t x i - t . x a . ' " . ^ ) | P 1/p . 2 — _ | p d x ) dt - 2 i f o l Y i . * ' " f : L p ( G n , n / 2 ) I I + ±- P f |Y± k l J h i ( h . - t J ^ M l t ^ d t . (p-1)! i - 0 ** 0 (*) By Minkowski's i n e q u a l i t y f o r i n t e g r a l s , ( | |; / K ( u , v ) d u [ p d v ) 1 / P « /( / | K ( u , v ) | P d v ) 1 / P d u , which w i l l , he used again i n Chapter F i v e . For a reference see Hardy, L i t t l e w o o d , and P o l y a , [,3l, #202;' p. lUT. v i t h the l a s t i n e q u a l i t y by the Holder c o n d i t i o n i n the d e f i n i t i o n of H r (0 ,M). p , x i n» Thus we have 2k where p/2 c 3,n " 2 m a x I l Y i J c l k i=0 . p/2 h a c, s m a x _ 1 _ I \y | / ( h i - t ) | t | dt. * k ( p - l ) t i=0 1 , K 0 From t h i s we conclude ( l ) f o r even k since I I * 5 V W I i l l * ! y G n , n / 2 > H -Now i f k i s odd, we suppose ( x 1 , * * * , x n ) e G n ^ 2 » h K n / 2 , and apply the usual T a y l o r formula: 3 k ~ 1 f ( x , t h , x 2 , , " , x n ) a*"3* 3 k f kH a + h~ T + h 3 k + 1 f ( x , + t , x t " ' t x T l ) + / (h-t) . d t . 0 3x* Using (7)» we i n t e g r a t e (8) over G Q and obtain 3 k f 3 k" Xf * / (h-t) [ / L-L2! » fP dx ]1/pdt c l 0 Gn,n/2 9xf h 2 i ( c 3 , n l | f : L p ( G n ) | | • c ^ M ) C 2 12 Thus we conclude t h a t & kf ix^ and we have completed the proof f o r p even. I f p i s odd, we subtract (U) from (2) and proceed analogously. Now we s t a t e and prove the extension theorem, f i r s t n o t i n g t h a t f o r any $ e L p ( G n ) , (/ U ( x ) | P d x ) l / p < ( / U ( x ) | P d x ) 1 / P . Gn\n Gn,n -+ ^ THEOREM 1. I f the f u n c t i o n f belongs t o the c l a s s Hp(G n»M) and n •> 0, then there e x i s t s a f u n c t i o n tyt the extension o f f from G* > n t o R n, s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s : i ) ty i s defined on a l l of R n, and II* * L ptR n >l l < c j | f : V G n , n ) l l i (9) n i i ) <J> c o i n c i d e s w i t h f on G n n ; i i i ) ty e H*(M') where P M. i i c l , n l l f : W H + c 2 , n M i ; 1 = ^ 2 » " ' • n- ( 1 0 ) The constants n and c 2 j r i depend only on n, G n, and the r ^ ' s , w h i l e c depends only on G n and n« Before p r o v i n g t h i s theorem, we note t h a t (.9) and (lO) e s t a b l i s h a r" bound on what i s e s s e n t i a l l y the Hp* norm of ty i n terms o f the H p norm of f . ( c . f . (9) o f Chapter S i x . ) 13 PROOF. For every p o i n t o f the c l o s u r e , G n j n * o f Gn*n l e t w and :ui' he b a l l s v i t h centres at t h i s p o i n t and r a d i i n/3 and 2n/3 r e s p e c t i v e l y . Because G*^ i s compact, we may s e l e c t a f i n i t e c o l l e c t i o n of b a l l s wl» uo» from the w's such t h a t y u covers Xj* Let t h e i r centres d i = l » be P l t P . Let h ( t ) denote a f u n c t i o n having continuous d e r i v a t i v e s of order r > r ^ (k = 1, 2, n) and s a t i s f y i n g h ( t ) = 0 f o r t <_ 1; h ( t ) • 1 f o r t > 2. :Firthermore f o r P e R n define h± = h ^ P ) = h( & ) ( i - 1, 2, m) ( l i ) where PP^ i s the p o s i t i v e distance from P^ t o the given p o i n t P i n R n. Let H = 1 - h 1 h 2 * , , h m . We now make the convention t h a t f = 0 outside Gfi, and c l a i m t h a t the f u n c t i o n * = fH (12) s a t i s f i e s the c o n d i t i o n s o f the theorem. I f the p o i n t P belongs t o G* > n, then i t l i e s i n one of the b a l l s ^ and thus h ^ P ) = 0. Therefore H(P) = 1 and 4(P) = f ( P ) , t h i s h o l d i n g f o r a l l P e O ^ . We see t h a t i f u <_2n/3, then U ^ l C G n u whence i f P e O J^ and P L i s an a r b i t r a r y p o i n t o f the boundary, L, o f G n, then P P L >. PIPL - P^P >. n - 2n/3 = n/3. Thus every p o i n t P e G n - G n > u l i e s outside a l l the u^'s and f o r them, h^ = 1 f o r a l l i, and H = 0. Then not only i s H bounded and continuous on l i t R n together w i t h i t s p a r t i a l d e r i v a t i v e s t o the order r , hut H i s a l s o i d e n t i c a l l y zero on a s t r i p i n G o f p o s i t i v e width a d j o i n i n g the boundary o f GnT, I f 0 < h < p/2, where p < n/3, then outside G*^/ 2 a 1 1 o f $( xi»'* * »xn)» $(xi+htx.2*" ' »x n), and $ ( x 1 ? h , x 2 , * • * ,x n) are i d e n t i c a l l y zero. Furthermore there e x i s t s a constant K such t h a t H and i t s p a r t i a l d e r i v a t i v e s t o order ' r > r k (k = 1, 2, n) are bounded above by K. Thus f o r x^ we have I = ( / | # ( P l ) ( x . 1 + h , x . . " ' f x n ) - 2*(Pl)Cx1,'*-,xn) + R° x l ^ x l + $ ( P l ) ( x 1 - h , x 2 , - . - , x n ) | P d x ) l / p x l - ( / . (fc1') f f ^ , ( x 1 + h , x 2 ,— ^ n ) H ( ^ > ( x 1 + h , x 2 , - - x B ) X P / 2 > = n 1 - 2 f ( ^ ( x 1 , x 2 , ' - - , x n ) H ( P ^ k ) ( x 1 , x 2 , - " , x n ) + + f ^ \ x1- h , x 2 , . . . , x n ) H ^ l - k ^ x 1 - h , x 2 , - - - , x n ) ] | P d x ) 1 / P - ( / , I I I < f x '(x + h , x 2 , - " , x n ) -G n , 12 k=° 1 - afi k )(x 1,x 2,--.xn) + f [ k ) ( x -h,x • • • , x n ) ) H ( p l " k ) ( x 1 + h , - " , x n ) 1 c x l 1 • x / ( p l - k ) , v + 2 f X i ( x 1 , x 2 , " - , x n ) ( H v ^ " ' ( x ^ h ^ , - ^ ) -(Pl "k )/ x * \ - H ^  ( x 1 , x 2 , " * , x n ) ) - f ( x ^ h . X g , * " ^ ) x ( x ^ ' j £ ) ( V h , X 2 , " , , X n ) " H ( x J " k ) ( x l - h » X 2 « ' " , X n ) ) ] P d x ) l / P < I <pi> t / I' (Pi-k) l P  k=° Gn,U/2 1 1 2 * 'fXl ( Vh , X2» " , » : S i ) " 2fX;L (xi»V'"'xn) + + f ( x 1 - h , x 2 , " , t x n ) r <3x ) + + 2( / | f x ( x l f x 2 , ' " , x n ) | x Gn,u/2 1 | H ( x ^ ) ( x 1 + h , x 2 , - , x n ) - H ( x l " k ) ( x 1 , x 2 , . . - , x n ; + ( J | f x k ) ( x 1 - h , x 2 , - " , x n ) | P | H ( x l - k ) ( x l * h 1  Gn,8/2 X . H ( x l - k ) ( x 1 - h , x 2 , . . . , x n ) | P d x ) l / P ] =0 Jk), v / L "» j I ^ v ^ X i + N » X 2 » *xn^  ~ t = 0 Gn,U/2 1 - 2 f (x 1,X 2,** #,X n) + f (x^h.Xg,'*'^)! db + 2h ( / |f(*)(x1,xa,-".xn)|P x Gn,p/2 1 (p.V-k+1) p 1/p x|H ^  (x 1+eh,x 2,*",x n)| dx ) + 2h( J |f x ( x ^ h . X g , " ' ^ ) ! x Gn,y/2 1 x|H ( Pl- k + l )(x, +Bh,x 2,...,x n)| Pdx ) 1 / P ] x^ x 11 16 < * X (I1) [ ( / |f^(x 1 +h,x 2,"-,x n :), -k=0 Gn,u/2 X l - 2 f ( ^ ( x 1 , x 2 ,— fx n) + f ( ^ ( x 1 - h . x 2 , — , x n ) | P dx) l / P + , 0 0 , + 2h ( / |f ( k )(x 1,x ?,"-,x )| P dx) l / P • r x i 1 ^  n * 2h ( J |f ( k )( X l-h,x p,"-,x )| P dx) 1 / P ] °n,M/2 X l 1K O i J l/Xl+h f ( x + l )(u,x 2,...,x n)du| P dx) l / p k=° Gn,p/2 *1 X l t ( / I/"1 f{l+l)Ut,2t'"^) du|P dx)l / P] • M j h l0 1 + Gn,u/2 x l ~ h 1 + 2pl+2h<cl,y/2Nf ; Lp(Gn)M • c2>u/2Ml) ), al the last inequality following from Lemma 1, where the term Mjjhl comes by setting k = p^ in the summation. Now let q satisfy l/p + 1/q = 1 and apply Holder's inequality to get >-« k r v x, '2' k~° °n,y/2 X l 1 t / - J f 1 |f ( f l )(u,x 2,...,x n)| P dudx^-dxj 1 7 1 5 ] Gn,p/2 x l " h 1 a1 PX+2 • Mjhl + 2 h(c 1 > y / 2||f : Lp(Gn)|| + c ^ ^ ) ). Now change the order of integration of x.. and u with 0 .< h < \i/k: I < K ( I (j^1) 2h **. /q-[l"'L \. ,if x - Cu,X2,v-,xnH dudx2'"dxn) ] k=o - •- * Gn>u/U- 1 17 i a l p l + 2 M l | h | + 2 ( c W 2 l | f : L p(G n)|| + c 2 f U / 2 M l ) ). Again using Lemma 1, P l - 1 1 ~ K ( J n ^ 2 h ( C l , M A M f : V G n ) M + C2.UA^ ) + • M^hl0"1 * 2 1 h (c1(„/2Mf : V n ' H * ^ / A ' ' (13) The last inequality ends the computation (for for 0 < h < u/U where we note that this calculation depends on n f i r s t setting p = n/U. Now for h > u/l+ we have (p,) ( P l) ( / |« • L(x 1+h,x 2,«",x n) - 2$ x (x l fx ,'--,xn) + R x-i^  1 (Pi ) n i /TI (P-I ) n ^ (x rh,x 2,---,x n)| P d x ) 1 / P < : )M <*|| ? (J1) ^ H ( p l" k ) : L ( < )|| Pi+2 /,} M ) a, < 2 1 (c X ; | | f : L (C )|| + Cp M.) K (Uh/ji) \ — i ,n P n ,r) x the last inequality following again from Lemma 1. Now set u = r\/k and note that we can increase the values of the constants C i ^ and ci 1 ^ so that equation (13) w i l l be satisfied for these values of x»n 2, n h as well. These results hold for a l l x^. Thus let (i) c l , n * m a x c i , n (i): c 2 t n = m a x c 2 n » i * and we conclude (10). The remaining part of the Theorem f o l l o w s immediately from (.12) i f we set c = maxlHJ G n si n c e the f u n c t i o n H depends on n. 19 CHAPTER FOUR ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE This chapter stands somewhat apart from the remainder of t h i s work because w h i l e these r e s u l t s f o r e n t i r e f u n c t i o n s belonging t o L p ( R n ) w i l l be used here s o l e l y f o r computational ease f o r p r o v i n g the imbedding theorem, the r e s u l t s are q u i t e i n t e r e s t i n g i n themselves. Theorem U i s p a r t i c u l a r l y s i g n i f i c a n t because i t gives L p i n c l u s i o n i n the d i r -e c t i o n opposite t o that which one normally expects: namely, i f g i s an e n t i r e f u n c t i o n of the e x p o n e n t i a l t y p e , 1 <_ p <_ p* and g e L p ( R ) t then g e L p , ( R n ) . Furthermore, Chapter F i v e demonstrates an i n t e r e s t i n g r e l a t i o n s h i p between the c o n d i t i o n s of d i f f e r e n t i a b i l i t y f o r f u n c t i o n s .of a r e a l v a r i a b l e and a n a l i t i c i t y f o r f u n c t i o n s o f a complex v a r i a b l e . Throughout t h i s Chapter and i n subsequent ones we speak of an e n t i r e f u n c t i o n , g, belonging t o the space L p ( R n ) . By t h i s we mean th a t as a f u n c t i o n of n complex v a r i a b l e s z^, Zg, **', Z r , g i s an e n t i r e f u n c t i o n on C n, w h i l e as a f u n c t i o n of n r e a l v a r i a b l e s , x^, x 2 , *"» XJJ, g belongs t o L ( R n ) . P We may consider the i n c l u s i o n i n the space L (R n) of e n t i r e f u n c t i o n s v vl» v2»*** n 1»?2»* n of n complex v a r i a b l e s z^, z 2 , " * » z n ot e x p o n e n t i a l types v i » v 2 » * " * v n r e l a t i v e t o these v a r i a b l e s . Such a f u n c t i o n g^ has the f o l l o w i n g pro-p e r t i e s : (see A h k i e z e r , [ l ] , pp. lfh-119) i ) i t can be expanded i n the power s e r i e s 20 x n a b s o l u t e l y convergent f o r a l l complex z^, ***» Z Q ; i i ) f o r any e > 0 there e x i s t s an A > 0 so th a t f o r a l l complex , * * *, z n we have | g t ( z 1 , - " , z n ) | < A e 1 i i i ) as a f u n c t i o n o f the r e a l v a r i a b l e s x-^, x n , g^ e L p ( R n ) . LEMMA 1. I f f ( z ) i s an e n t i r e f u n c t i o n o f exponential type v, and f belongs t o L (R), f o r some p > 1, then f o r a l l r e a l x and any a E R, s i n . a f ' ( x ) - vcos a f ( x ) = £ , , k - l s i n 2 a kir-a = I (_i) f ( + x ) . k=-~ ( a - k i r ) ^ v We comment t h a t our a c t u a l concern here i s t h a t f be bounded on the r e a l a x i s , and t h i s i s i n f a c t the case as w i l l be seen from the proof o f t h i s lemma and as a consequence o f Theorem U of t h i s Chapter. f ( z ) - f ( 0 ) Then we co n s t r u c t a f u n c t i o n g(z) = — — — — — — — which belongs t o z Lg(R) , and from t h a t the i n t e r p o l a t i o n formula ( l ) i s e s t a b l i s h e d . PROOF. Let g(z) be defined by f ( z ) - f ( 0 ) g(z) = whence g i s an e n t i r e f u n c t i o n of type l e s s than or equal t o v. To show that g belongs t o Lv>(R) i n the case p > 2, we apply the HSlder i n -equa l i t y . I f 1 <_ p <_ 2 we argue as follows. F i r s t i f f e L-^R), then the A F o u r i e r transform of f , f , belongs t o L (R) and by the Wiener-Paley A A Theorem, f has compact support. But t h i s implies that f i s i n L^(R) and thus f belongs t o L W ( R ) , i . e . f i s bounded. S i m i l a r l y i f f e L 2(R) then f e Lg(R) and again has compact support, whence f e L ^ R ) . For the case A 1 < p < 2 we apply the Hausdorff-Young Theorem to conclude that f e L^i(F.) ( l / p + 1/p* = l ) , f has compact support and again f belongs to Loo(R). In a l l cases f i s seen t o be bounded, whence we conclude that 2 / |g(x)| dx < «. Then on the basis of the Wiener-Paley Theorem again, there e x i s t s a function * e L 2(-v,v) f or which / \ 1 rV i z u / \ g(z; = — J e *(u)du 2TT -v or z , v i z u f ( z ) = f(0) + — / e $(u)du 2TT -v or i n s t i l l another form, f ( z ) = f(0) - — JV $(u)i-(eiZU)du. 2ir -v 8u We are i n t e r e s t e d i n the expression U(x) = sinct f'(x) - vcos a f(x) 22 which may be expanded i n the form U(x) = -vcos a f(0) + 1 , V g + — , / *(u) — ( e ( u s i n a + i v c o s a) ) du 2ir -v 3u We now decomtiose the f u n c t i o n V(u) = - i e ^ a U ^ ( u s i n a + ivcosoi) (-v < u < v) i n F o u r i e r s e r i e s which takes the form V(u) = v s i n a \ (-1) g e k=-°° (a-kir) I n t r o d u c i n g t h i s i n t o the expression f o r the f u n c t i o n U(x) and applying (3) we get ivsi n ^ c t U(x) = -vcos a f(0) + x 2ir r V , ,? ( " l ) k i u ( ( k r a ) / v + x ) . x J »( u)— (I 2 e )du -v 3ft -00 (a-kir) 2 = -vcos a f(0) - v s i n a x « (-I)' . ) k x J -r? 2 ( f(kir-a/v +x) - f(0)) - 0 0 (a-kir) 2 £ = v s i n a £ 2 f(kir-a/v +x) -» (a-kir) w i t h the l a s t i n e q u a l i t y h o l d i n g since 0 0 ( r l ) k d 1 c o s a 1 5" = = ~ * ka=-<» (a-kir) da s i n a s i n a This proves ( l ) . For a complete d i s c u s s i o n of t h i s i n t r i g u i n g i n t e r p o l a t i o n f o r -mula, we r e f e r t o Ah k i e z e r , [ l ] , pp. 182-189. As a source f o r the Wiener-Paley Theorem we r e f e r t o Zygmund, [l8], pp. 272-27^ , and f o r the Hausdorff-Young Theorem, [l8], pp. 25U-258. The theorems of t h i s chapter are p r i m a r i l y based on the f o l l o w i n g lemma which i s an immediate consequence o f Lemma 1. LEMMA 2. I f the f u n c t i o n $ ( x 2 , " * , x n ) i s , r e l a t i v e to x-^, an e n t i r e f u n c t i o n of type v f o r almost a l l ( x p , " * , x ) and <J> e L ( F r ) , then P ix± belongs t o and 3* I I— : L (R n) || £ v | | * : L ( R n ) | | . dx± p p PROOF. As a f u n c t i o n of x^, $ s a t i s f i e s the co n d i t i o n s f o r f i n Lemma 1. Hence s e t t i n g a - ir/2, we ob t a i n = -n- I - i " 1 ^ 9 $ U ( k - — ) / v +x, , x 9 , " * , x n ) . 3x ± ir k=-» ( k - ? l / 2 r Thus | | ^ - : L ( R n ) | | <vjf _ i _ 2 | | * : L ( R n ) | | 3x x p k=-» ( I - 1/27* p = v|| * : L p ( R n ) | | V 1 since l * - 1. This completes the proof. k=-» (k- 1/27 ir From Lemma 2 we have, i n p a r t i c u l a r , t h a t i f g^ i s an e n t i r e f u n c t i o n of types v^, v n , and g^. e L p ( R n ) , then I h S : L ( R n ) | | < v ||g. : L ( R n ) | | . 3 ^ P k v p The f o l l o w i n g three theorems are r o u t i n e , but w i l l l e a d t o the more s i g n i f i c a n t r e s u l t s o f Theorems k and 5. THEOREM 1. Let g v ( z ) belong t o L p ( R ) and be an e n t i r e f u n c t i o n o f ex-p o n e n t i a l type v; l e t a > 0, h = a/v, x k = kh (k = 0, £L, +2, • • • ) , and i k pa/p l e t x k s a t i s f y x k <_ x' <_ x^ + rh where r i s an i n t e g e r . Then ( h I |g ( x ^ l P ) i / P < (1 + r«)||gv : L ( R n ) | | . PROOF. C l e a r l y 0 0 t) 0 0 Xk+1 » r, / | g v ( x ) | P d x * I J |g ( x ) | p d x = h l |g ( t ) | P _» ~~ x k where x^ <_ 5 k ^ x^+i* Furthermore, using Lemma 2 and s e t t i n g 1/p + 1/q we have by Holder's i n e q u a l i t y »1 / pi < ! k ( < ) i p ) ^ . ( ! i g v ( ? k ) i p ) 1 / p i k=-m k=-m <»1/pcj -gv<VlV/p k=—m 1/p m x k + l . - , <V t i (/ !g;(t)idt)p}1/p k=-m x^  , . _ x,,+rh 1/p. m t K i '/ \iP , xP/<l,l/P lh [ I j |gv(t)|pdt (rh)p/H] k=-m ^ <h(r1+P/qf |g'(t)|Pdt)1/p * rh||g^  : L p ( R ) | | < ra||gv : Lp(R)||. 25 Therefore m h 1 / p < i i g ( x ; ) i p ' 1 / p k=-m K v.1/p r ? I / * \ I p r , , t %|Pil/P A vl/p, ? I , T M P X 1 / P = h I i | g ( x ^ ) r - 1 |gv(5k)n • h ( 1 lg/vl ) k=-m k=-m k<=-m < (1 + r o ) | | g v : L ( R n ) | | . L e t t i n g m -»• ve obtain (5) and the proof i s complete. THEOREM 2. Let g-*- belong t o the space L (R n) and be an e n t i r e f u n c t i o n V P of types v l f v n ; l e t < 0, h k = c ^ A ^ (k = l,*«« tn), x k = s h k (s = 0, ±1, +2, *•*) and l e t x s a t i s f y x* < x < x + rh. where r i s • — » — > s ' S — S — S K an i n t e g e r . Then we have the i n e q u a l i t y sup [ ( I k ) I |fo(x* - u , . — - u ) | P ] 1 / P u A k = l * -~ — v s l 1 s n n n <. H (1 + r a )||g* : L (R ) | | , k=l x v p where the sup extends t o a l l r e a l u ± , ***» and the summation i s c a r r i e d out on the i n t e g e r s s^,»**, s n . The proof o f Theorem 2 i s by i n d u c t i o n from Theorem 1. We omit the d e t a i l s . THEOREM 3. Let g^ be an e n t i r e f u n c t i o n o f types vx»***» v n f o r which the _k l e f t hand s i d e o f (6) i s f i n i t e ; l e t a, . h, . and x s a t i s f y the co n d i t i o n s k k s of Theorem 2. Then g^. belongs t o L ( R 1 1 ) . and °^ CT p (6) Mg- : L (RN)|| ! [ ( 2 r ) n S h . ] 1 / P x x 8Up( J — I Ig^x"1 -u^—.x* -un)(P)1/P 11 CD 00 V S i S_ u. -*° -°° v s l "n PROOF. We must show t h a t ||g+ : L^CR11) | | i s f i n i t e . eo oo J •••/ |g^(x 1,*",x n)| Pdx 1«"dx n oo Tl ^ = I'" I / J 1 , - J J N |g^ 1,-,x n)|V-dx n -°° - 0 0 X -1 X -1 S l s 0 0 » 2 r h n 2rh , _ -» -» 0 0 1 n 2rh 2rh 0 0 0 0 i p </ •••/ I'" I |g*(x" - u - u )| dun'-'du 0 o -» -» v s l 1 s n n 1 n < ( 2 r ) n ( ni^) sup J— I |g*(xt -V",*^ -uDp. k=l -°° _oo 1 n Therefore ||g^ : L p ( R n ) | | i s f i n i t e , and we conclude (7) a f t e r r a i s i n g t o the power 1/p. Tl/v We note t h a t the constant (2r) may be omitted i n (7) w i t h the f o l l o w i n g v a r i a t i o n : i f the p o i n t s o f i n t e r p o l a t i o n x are e q u a l l y s k spaced, t h a t i s i f we replace x by x i n ( 7 ) , then i t would be s u f -S k s k h i 2 r h i f i c i e n t t o take the i n t e g r a l J f o r approximation i n s t e a d o f J 0 0 i n the t h i r d l i n e of the proof. This would e l i m i n a t e the constant n/p (2r) , and i n e q u a l i t y (7) hecomes Me* : V R n ) ! l < (m,/'* s u P ( f . . . ll^x1 . U l , - . , x n - u n ) | p ) 1 / P . -» -«» 1 n E x p l i c i t mention may now be made of the r e l a t i o n s h i p s between i n e q u a l i t i e s (6) and ( 7 1 ) . I f we l e t r = 1 i n (6) and define ((g+ : L„(R n) )) 00 00 = sup [ ( n h k ) X-.. J l g ^ C x 1 - u n ) f ) 1 / p U j -» -» s l n as i n Theorem 2, we then have the i n e q u a l i t y ||gg : L p ( R n ) | | < C(g* : L p ( R n ) )) < n ( l + c^Ng* r L p ( R n ) | | . LEMMA 3. F6r 1 ^  p <_ p' < » and f o r a l l non-negative numbers al» * * *» '"n w e n a v e ( X \ ) i ( X \ > • k=l K k=l K Using Theorems 1,2,3 and Lemmas 1,2 we now prove the s i g n i f i c a n t r e s u l t s of t h i s Chapter which give ( f o r e n i t r e f u n c t i o n s o f the expon-e n t i a l type) L (]Rn) i n c l u s i o n f o r d i f f e r e n t values o f p and d i f f e r e n t dimensions of R11. These two theorems w i l l be used t o prove the major r e s u l t o f t h i s work which i s the imbedding theorem o f Chapter 7. THEOREM h. I f the f u n c t i o n S ^ i s an e n t i r e f u n c t i o n o f exp o n e n t i a l types y., ***, u and belongs t o L ( R n ) , and i f 1 <_p < p" <_» , then g ^belongs t o L p i ( R n ) , and 28 18+ *V(Rn>ll l 2 " (kV*)1/P " 1 / P' Ng- : VRn)U-PROOF. From the v a l i d i t y o f (8) and Lemma 3, < (; ( i — i - U n ) ip' >l/p' k=l u i -°° -« 1 s n n -i /_» co oo n 1/r) < (n ^ r 1 5 sup ( i — i \^(x] -nlt~\*ns ) k=l u^ -» -» v s l n n l/t) - l / n ' n < ( H h.) n (1 + a j | | f o t : L ( R n ) | | k=l ^ k=l ^ v P = n (1 • o^) (vk\)1/V " ^ ||g- : L ( R n ) | | , k=l which holds f o r any > 0 (k = 1, 2, *•*, n ) . Let u = l / p - l / p ' . The f u n c t i o n ( l + a)/aw a t t a i n s i t s miniraun (on the i n t e r v a l (0,l) ) o f (10) < 2. (11) U)» x l - W — CO (1 - to) Therefore i t f o l l o w s that n i fn - 1 lr>' l|g+ : L p ( R n ) | | <2n ( n v k ) \ \ H : L p ( R n ) | | <». Then g-> e L i ( R n ) and we have completed the proof, v p THEOREM 5. I f g^ i s an e n t i r e f u n c t i o n o f the exponential type belonging t o Li (R n) and i f 1 < m < n, then f o r a l l f i x e d x , , **•, x i s P — m+l' * n B v 29 i n t e g r a b l e i n the p-th power, and n *t ly\ lie*: y ^ w ^2°( n V N « $ : L P ( R N ) H - ( I 2 ^ m+1 PROOF. From (8) we have t h a t n (1 + ^Hg^ : L ( R n ) | | k=l v y > (n h k ) l / P s u p [ I .-I Ig^x* - u ^ ' - . x " - « n ) | p ] 1 / p 1 " i s l s n 1 n > (H h^)1/p.«p sup [ I ' " I Ig^x 1 -nlt"\xns - u n ) | P ] 1 / p 1 uj s m + 1 / . . , s n 8 l sffi s l n > (n h , ) 1 ^ sup [ I - I : f g ^ - u , , " . ^ • ^ ^ 1 . - ^ ) | p l 1 / p 1 V ' . U m s i sm 1 m £(n h k) 1 / P[(n h k ) 1 / p sup ...I !g^(xj ^ . — j? - u m , - * * , x n ) | P ) 1 / p ] a f t l 1 V " * ^ s l sm 1 which holds f o r a l l a > 0. We l e t a. = 1 and apply (8) t o g-> as a k K V f u n c t i o n o f m v a r i a b l e s : 2n|.|g* : L p ( P n ) | | > ( nhk)l/P||gj : L p ( R m * m+1 and we conclude (12) n o t i n g t h a t h^ = a k / v k ' 30 CHAPTER FIVE APPROXIMATION OF CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE We now begin the technique of qpproximation of continuous func t i o n s by e n t i r e f u n c t i o n s o f a complex v a r i a b l e . While there are other methods of t r e a t i n g the continuous f u n c t i o n s (e.g. d e a l i n g only w i t h those f u n c t i o n s o f compact support and then e x t r a p o l a t i n g by " d e n s i t y " ) t h i s approximation allows us t o c a l c u l a t e the "hard core" i n e q u a l i t i e s u s i n g the convient e n t i r e f u n c t i o n s r a t h e r than the continuous ones. Let f e LptR 1 1). We may consider a l l p o s s i b l e f u n c t i o n s g^x^,* • • ,x n) a l s o belonging t o L (R 1 1) and having the property t h a t f o r almost a l l (x2»"*,x n) they are e n t i r e f u n c t i o n s o f ex p o n e n t i a l type v w i t h respect t o x^. As a measure of how c l o s e l y the f u n c t i o n f can be approximated by such e n t i r e f u n c t i o n s g v , we define the q u a n t i t y : where the infimum Is extended over a l l p o s s i b l e g y . LEMMA 1. I f the f u n c t i o n f belongs t o L p ( R n ) and f has p a r t i a l d e r i v a t i v e s 3 P f — a l s o belonging t o L (R ), then f o r a l l i n t e g r a l s >^0, a x ^ P | | A ( x + 8 ) ( f , h ) : Lp(R n ) I| < h P||A S ( f ( P ) , h ) : L ( R n ) | | . (2> PROOF. I t s u f f i c e s t o show (2) f o r p = 1 and s = 0. |A ( f , h ) : L ( R n ) | | = (/-«/ |A ( f , h ) | P d x 1 . . . d x n ) 1 / p eo oo x +h / - (/ — / I / 1 * i ( t , x 2 , - " , x n ) d t | P d x 1 . . . d X n ) 1 / P - 0 9 -» x x 1 < (/" -~f\tx | P d t h P / < 3 d x ^ - d x / ^ —oo _oo 1 l / p + 1/q . . r°° . i ,p . l / p <h (/ "'J | f | dx ••*dx n) —CO _oo x l -h I If : VR n )M x l p where l / p + 1/q = 1. Thus | | A ( X + S ) ( f , h ) : L (R n) | | i h M A ^ - ^ C f * ,h) : L (R n) 1 p x l x l p , P i i , s , (p) < — < h P||A S ( f V P / : L ( R n ) | | . x ± x^ p We now e s t a b l i s h the c r u c i a l s u f f i c i e n t c o n d i t i o n s f o r the approx-imation by e n t i r e f u n c t i o n s . Note t h a t i n t h i s d i r e c t proof we a c t u a l l y e x h i b i t an e n t i r e f u n c t i o n which approximates f t o w i t h i n a s p e c i f i e d d i s t a n c e . THEOREM 1. For a l l r > 0 and a l l v, there e x i s t s an e n t i r e f u n c t i o n ( r ) K (u) of exp o n e n t i a l type v s a t i s f y i n g the f o l l o w i n g three c o n d i t i o n s : i ) /" K ( r ) ( u ) d u = 1 i i ) f" |K ( r )(u)|i.du < a„ \\ • r i i i ) f o r a l l f u n c t i o n s f "belonging t o H r (M) we have t h a t P» xl oo r \ | [ f - J K ^ (u- X l)f(u,x 2,"»,x n)du : L (R 1 1) j | <^  b ^ / v —00 P where the constants a r , b r depend on r but not on v. From (5) we con-clude t h a t A Cf) < b_M/v v , x x p - r f o r a l l f e H r (M). P.x x PROOF. Let g(z) he an e n t i r e f u n c t i o n o f type u n i t y s a t i s f y i n g the co n d i t i o n s 00 i ) / g ( t ) d t = 1, _oo oo i i ) J j g ( t ) l d t < », oo i i i ) b r = / | g ( t ) | | t | F d t < oo. —CO We f i r s t note t h a t , vp+l P+2, , P+ 2 , x i - l ,p+2, , (-if ' A (*,h) = I (-1) ( P. )*(x + i h ) i=0 1 * I p. *(x + i h ) - $(x) i = l , . i - 1 ,p+2. p t 2 where p. = (-1) ( . ). Then £ p = 1 . 1 i = l 1 Now l e t f e H (M). Consider the f u n c t i o n defined by P.*! 33 g v ( X l , - - \ x n ) - f g ( t ) [ ( - l ) P + 1 4 P + 2 ) ( f , t / v ) + f ( X l , - - - , x n ) ] d t — 0 0 1 «• p+2 = / g ( t ) I P ^ ^ + t / v , x 2 , ' - ' , x )dt _» i = l 0 0 ( r ) = / K ( t - x 1 ) f ( t , x 2 , " ' , x n ) d t where ( \ P+2 K ' = A P i V / 1 g(vu/i)* ( r ) Then K y i s an e n t i r e f u n c t i o n o f type v , and hence so i s g^ w i t h respect t o x^. Moreover, g - f = ( - 1 ) P + J g ( t ) A P + 2 ( f , t / v ) d t , -oo 1 -» Pr 1 Thus from (2) and from f's membership i n H r we have P » xi ||g - f : L ( R n ) | | < M/v P + C t / | g ( t ) | | t | P + a d t = b rM/v r (12) " —oo which proves ( 5 ) . From ( 7 ) , ( 1 0 ) , and ( l l ) we conclude ( 3 ) . Furthermore oo , \ p+2 oo J |K* r'(u)|du< I |p | J v / i |g(vu/i)|du —oo V 1 —oo P+2 oo = I I P J / | g(u)|du = a r , and the proof i s complete. 3U We now e s t a b l i s h s i m i l a r c o n d i t i o n s f o r the approximation o f fun c t i o n s belonging t o the c l a s s H p(M) by e n t i r e f u n c t i o n s . We note th a t the q u a n t i t y A + ( f ) p = A v ••• v ^ f^p i s S^^ 1 1 ^ Anj(f) = i n f | | f - g-+ : L ( R n ) | | (13) v P g v v p where the infijBum i s taken over a l l e n t i r e f u n c t i o n s g^ of exp o n e n t i a l types Vjj. in the variables x^, k = l t * * * , n . THEOREM 2 . I f the f u n c t i o n f belongs t o the c l a s s Hp(M"), then * k=l v. K k where d depends only on r ± , ***, r R . We give an o u t l i n e of the proof f o r general n. For a f u l l d i s -cussion o f the case n = 3 we r e f e r t o [ 9 ] , pp. 2 5 9 - 2 6 0 or pp. 17-19 i n the t r a n s l a t i o n . (r±) PROOF. Let K be def i n e d as i n ( l l ) f o r i = 1 , n. Then , ( r l > ( r n } FT 1 n x ftx^u^Xg+Ug,*" .Xjj+Ujjjdu, and ( r 2 ) ( r ) R I n n - f (x,+u, ,x 2+u 2,• • * ,x +u ) ]du = l\\ k=l K where the ifi^'s are defined as f o l l o w s : 35 ( r . ) ( r ) *1 = / n K v v n <V t t t x ^ r " , * * ) -R 1 n f(x 1+u 1,x 2V* ,,Xn) ^ d u - ( r . ) = / K v ( u i ) [ f - ( X ; i y , x n ) - f t x ^ . x g / " ^ ) ] du x * 2 - / K ^ ^ u ^ ' - ' K v n ( u n ) [ f t x ^ . X g , — ^ ) -R*^  1 n f ( x x + u x ,x2+U2 ,x 3,' *' ,x n) ] du 0 0 ( r , ) = J K v ( u x ) h 2 ( x 1 + u 1 , x 2 , - " , x n ) d u 1 ( r . ) ( r ) »i - v x K^"* v n (un) [ f ( x l + t t l » , , , ' x U + n i - l ' x l » , M ' x n i " - f ( x 1 + u 1 , " , , x . + u i , x i + 1 , " ' , x n ) ] du 0 0 ( r , ) ( r , ) ~ J — / K 1 ( V - K / - 1 ( U ^ ) x -co _» V l i - 1 x h± ( x x + u x , • • • ^ , • * * ,x n) du x* * • d u j . ^ where h i < xl + ui•*''. xi-l + ui-l» xi • * *'» xn } (rj = K v. (u.) [ f C x ^ . - " ^ ^ ^ , - - - ^ ) -- f(x x+u 1,*",x i+u i,x i + 1,***,x n) ] . Then by (5) Mhi : Lp^n)M l ^ V ^ and II*! : VRn)H < K*!^*1 » i i*i *y*n>n <f ' i V ^ v - ^ ^ V . ,)i * * _eo -«> 1 i - 1 x (/ n l hi (V^»* ,^ xi-l +Vl» xi»'''» xnH P t o) 1 / P dV ,' d^ < a a ---a ||h : L ( R n ) | | r l r 2 r i - l 1 p < a (b^M./v. 1" 1), ( i = 2, n ) F i n a l l y we conclude t h a t n f - f o : L ( P n ) | | < I IU k : L ( R n ) | | k=l v b M x b r M 2 b M T /1 r l v / 2 r r n r n i d n » M k which proves ( l U ) . The f o l l w i n g t e c h n i c a l lemma i s s t a t e d here without proof. A proof may be found i n [ 9 ] , pp. 2 6 1 - 2 6 2 , or pp. 1 9 - 2 1 i n the t r a n s l a t i o n . 3 7 LEMMA 2. Let ( f ^ ) he a sequence o f fu n c t i o n s each o f which belongs t o L ( R n ) and has p a r t i a l d e r i v a t i v e s p a l s o belonging t o L (R 1 1). I f | | f - f k : L p ( R n ) | | * 0 « s k + -and p 3 f $ - a : L (R")| | •»• 0 as k f » 3x, P p v l then f may be mo d i f i e d on a set o f measure zero so th a t the f u n c t i o n 3 P f f has the p a r t i a l d e r i v a t i v e ~TT s a t i s f y i n g dx^ 9 P f — = 3 x l We now e s t a b l i s h a converse t o Theorem 1 , showing t h a t a c o n t i n -uous f u n c t i o n which can be s u i t a b l y approximated by e n t i r e f u n c t i o n s r a c t u a l l y belongs t o the space H p THEOREM 3 . Let r >. 0 , f e Lp(R n) and suppose t h a t A " 1 K/\>r (16) k f o r a l l v running through a geometric progression a ( k = 0 , 1 , * * * ) w i t h commom r a t i o a > 1 . /Fjurthermore, l e t g designate an e n t i r e f u n c t i o n , w i t h respect x^ of exponential type u n i t y f o r which Wt - g : Lp(Rn)ll I K . ( 1 7 ) Then the f u n c t i o n f, = f - g belongs t o the c l a s s H1* (M) where M - CjJC w i t h c r depending on r but not on K. PROOF. Let g v = gv(xlt'* *,x R) designate an e n t i r e f u n c t i o n o f type i n f o r which | | f - g v : y R 1 1 ) ! ! < K/v J where we make the convention t h a t v = a k = m k (k = 0, 1, • • • ) . Then because f s a t i s f i e s the c o n d i t i o n s o f the theorem, i t may be re presented i n the form f • g + * where g = g m and o » k=l *"k m k - l (We note t h a t t h i s technique w i l l be used f r e q u e n t l y i n the sequel.) Now d i f f e r e n t i a t e t h i s s e r i e s p times w i t h respect t o x ± . Then x l k=l where 9 (gm k " Sm k_ ±) \ " F 9 x i From Lemma 2 of Chapter U and (16) r+1 n P 2K kp 2K a ! ||Qk : y R )|| < m^ — = a ( K J ) ( P + « J 1 ~ ^k—1 a a —ka Because the s e r i e s dominated by a c o n v e r g e s t h e s e r i e s (19) con-verges i n L p ( ^ n ) norm. Thus by the preceeding Lemma, the f u n c t i o n $ belongs t o L (R 1 1) and has a d e r i v a t i v e o f order p, and the s e r i e s (19) converges t o t h a t d e r i v a t i v e i n L (R n) norm. 39 Now l e t h > 0 and f i n d a natural number N f o r which . N + 1 n 1/a < h <_ l / a w , or equivalently N+1 K a > 1/h > a Then we have that |A* (* ( p\h) : L ( B n ) | | < x l x l P N 2 k = 1 1 P k=N+l K P Now from Lemma 2 of Chapter U and i n e q u a l i t y (2) of Lemma 1, we have that N 2 II|AX (Q k,h) : L p(R )|| 1 1 2 r , l r t(g) _ , n. , . 2 5 2k ± h : L ( R n ) | | < h 2 ^ 1 » xl P 1 (2-a)(N+1) 2-a ka a r+1 2 a ~ a ' a 1 0 1 2 = 5 — <20> a - 1 r+1 2 (2-a)(N+1) r+1 2 2-a a-2 < a K a < a ' Kh a h r+3 a < a Kh , and N+1 N+1 a Uo From t h i s we conclude the theorem by s e t t i n g r+h r+3 * i , . c r « a + — . (22) a -1 We make s p e c i a l n o t i c e here o f the equivalence o f the spaces def i n e d by the f i r s t o r second d i f f e r e n c e c o n d i t i o n on the p-th d e r i v -a t i v e o f f . Theorem 1 may be proved u s i n g only the f i r s t d i f f e r e n c e f o r a < 1. That i s , a f u n c t i o n s a t i s f y i n g the Holder c o n d i t i o n on the f i r s t d i f f e r e n c e o f the d e r i v a t i v e may be s u i t a b l y approximated by e n t i r e f u n c t i o n s . The c r u c i a l n e c e s s i t y f o r separation of the cases ct < 1 and a = 1 i s demonstrated i n the proof of Theorem 3 where the i n -e q u a l i t i e s (20) remain v a l i d u s i n g the f i r s t d i f f e r e n c e f o r «< 1, but f a i l t o h o l d on the f i r s t d i f f e r e n c e when c t = 1. Thus a f u n c t i o n approx-imated i n a c e r t a i n fashion by e n t i r e f u n c t i o n s w i l l s a t i s f y both the f i r s t and second d i f f e r e n c e s on i t s p-th d e r i v a t i v e i f a < 1, but only the second i f a = 1. Therefore when, i n the f o l l o w i n g theorem, we note t h a t t h i s approximation i s both a necessary and s u f f i c i e n t c o n d i t i o n f o r membership i n the H - c l a s s , we w i l l have demonstrated the equivalence f o r a < 1 of us i n g the f i r s t or second d i f f e r e n c e , w h i l e i t remains necessary t o r e t a i n the second d i f f e r e n c e when ot = 1. Now r e f e r i n g t o Theorem 1, we see t h a t the f u n c t i o n g o f expon-r e n t i a l type u n i t y w i t h respect t o x a l s o belongs t o the c l a s s H 1 P t x l but only w i t h a d i f f e r e n t constant depending on | | f : L ( R n ) | | . In t h i s case i t f o l l o w s from (IT) t h a t ||g : L p ( R n ) | | < | | f : I p ( R n ) | | + K (23) and from Lemma 2 of Chapter ht Us^ * V H n ) M < | | f : L p ( R n ) | | + K . Thus M ^ C s ^ . h ) : L p ( R n ) | | < k ( | | f : L p ( R n ) | | + K) and from Lemma 1 of t h i s chapter we have that MA*. ( g { x\h) : L (R n) | | x l x l p < h | | g ( p + l ) : L (R n)|| < h ( | | f : L_(R n)|| + K ) . x l " The l a s t two i n e q u a l i t i e s imply that f o r a l l h Combining t h i s with Theorem 5 of Chapter h and Theorem 1 of t h i s Chapter we have the p r i n c i p a l r e s u l t of t h i s Chapter which w i l l be the basis of a l l future c a l c u l a t i o n s . THEOREM k. In order that the function f e Lp(R n) belong to the class H*" _ ( i . e . H** (M) for some constant M) i t i s necessary and s u f f i c i e n t P » x x P * 2_ that there e x i s t s a constant K f o r which V ( f , p i I / , r f o r a l l v >1, or at least f o r a l l v running through a geometric pro-le gression v = a ( a > l , k = 0 , l , * * • ) . The s u f f i c i e n t condition follows from Theorem 3 by a d d i t i v i t y i n H r . P. x x c l a s s H p F i n a l l y we formulate r e s u l t s equivalent t o Theorem 3 f o r the r V n. R ) s a t i s f y n K, i where K^'s are constants, r ^ > 0, and the v^'s run through a geometric progression » a^ (k = 0, 1, *** ) w i t h a.. > 0. Let g he an e n t i r e f u n c t i o n o f type u n i t y i n each of the v a r i a b l e s such t h a t n n | | f - g : L (R )|| < I K.. P 1=1 - + Then the f u n c t i o n $ = f - g belongs t o the c l a s s H (M) where P n M. a C T* _ J i " u r X K i * 1 i .1=1 J PROOF!. R>r the approximation of type w i t h respect t o x 1 we obv i o u s l y have the i n e q u a l i t y n K. L. x ( f ) < A*(f)_ < J — r r v 1 » x 1 P - v p rx f o r any v_, * * *, v . Thus <£ n K l ^ V , v ( f ) P i — i - 1 7 -i * i vi v i I t f o l l o w s then from (27) and (26) t h a t , on the b a s i s o f Theorem 3, the f u n c t i o n $ as a f u n c t i o n o f the v a r i a b l e x-^  belongs t o the c l a s s r l H (M. ) where P.X-L 1 n 1 r l 1=1 1 The same argument holds f o r X 2 » " " » x n » m& the proof i s complete. 1»3 CHAPTER SIX THE NORMED SPACE H^ p With the technique o f approximation "by e n t i r e f u n c t i o n s we can now demonstrate many p r o p e r t i e s of c l a s s e s o f fun c t i o n s defined over n r R . In t h i s chapter we prove completeness o f the normed space H p and some p r e l i m i n a r y imbedding theorems. As s t a t e d b e f o r e , i n t h i s chapter we use the equivalent d e f i n i t i o n ? r f o r H • namely, a f u n c t i o n f belongs t o H (M) i f f has d e r i v a t i v e s P P» xj £ a l l belonging t o L p ( R ) f o r k = 0, 1, "*,o and i f ||Av ( f ( P ) , h ) : L ( R n ) | | < M(h|° (a < l ) \\L 2 ( f ( p ) , h ) : L ( R n ) | | < M|h| (a « l ) . x i x i p The f o l l o w i n g two lemmas l e a d t o the i n t e r e s t i n g Theorem 1 r which, i n t u r n , i s used t o prove completeness o f Hp. LEMMA 1. An e n t i r e f u n c t i o n g = g+(x x,* * * JXJJ) o f types i n the v a r -i a b l e x k which s a t i s f i e s ||g : L f R n ) | | < M belongs t o the c l a s s H (M ) where ' , r i M x = UMVj . PROOF. From Lemma 2 of Chapter U and Lemma 1 of Chapter 5, I*x ^ . h ) : L (R n ) | | < | | g ( p ; i + l ) : L (R n)|| |h| i f [v ±h| <_ 1. On the other hand, i f | V j h j > 1, " A * ( « < x i > ' h ) : VR n )" i 2M* ( P i ) : V*n>N x i X i P X j P 2Mv P i|v.h| a i = 2MvT i|h| a i. This concludes the proof f o r < 1. I f a x = 1 and i f |y.h| <_1 MA^g^.h) : L p ( H n ) | | <«vJPl+2)|h|2 <Mv^ |b| , whereas i f |v ih| > 1 " A x 1 ( g ( x l > ' h ) 5 V p n >M<Ml g ( x f : L p ( H n ) | | Pi r* <_ UMVi f.'tMVj |h|. LEMMA 2. Let the function f = f(x,,**',x„) be the sum i n L -norm of i * * n p the s e r i e s 00 fUi.-'-.Xn) « I Qs(xi,**-,x n) s=0 s/rjj where Q g (s=0, 1, •••) i s an enti r e function of types v k = 2 i n the variables x k which f u l f i l l s the i n e q u a l i t i e s llQo ' VRn)!! ±A M°s : V*"*!! <.B/2 8 (s = 1, 2, ••• ) where A, B, and a are a r b i t r a r y , not depending on s. Then f belongs t o the class H (K) where r x = ar^ and % = *** = K n = K<_UA + B. Furthermore, | | f : L p ( R n ) | | <_ A + cB where the constant c does not depend on A or B. PROOF. Let u-1 3=0 By the c o n d i t i o n s o f the theorem, S^ - i s an e n t i r e f u n c t i o n which i s o f expon e n t i a l type l e s s than u/r> v f c = 2 ( u = 1, 2, ) i n the v a r i a b l e x^. Thus from. (2) lk -s* » y* n )M i X I K : y * n ) l l s=u oo 1 1 < I B/2aS = B ff — < c B/2 a U s=y 1 - 2 2 M J n C3B ' a r k • k=l v. * k Let f = Q Q + *. By (U), A+(f) s a t i s f i e s the co n d i t i o n s o f Theorem 5 of the previous Chapter, and we conclude t h a t * belongs t o H (K ) P 1 i t 1 where = ar^ and = K w i t h K < c^B. F i n a l l y , Q i s an e n t i r e f u n c t i o n of type u n i t y i n each v a r i a b l e x, , s a t i s f y i n g ( 2 ) . Therefore from Lemma 1, 0n belongs t o the c l a s s H (K ) w i t h r as above and P K± = HA. Thus f l i e s i n the c l a s s (?) w i t h K± <_ hk + c^B. The second donelusion o f the theorem f o l l o w s d i r e c t l y from the i n e q u a l i t y I I f : L (RN)|| <_A + I B / 2 d S = A + cB P s=l and t h i s completes the proof. We now prove the f o l l o w i n g r a t h e r unusual theorem which s t a t e s t h a t c e r t a i n mixed p a r t i a l d e r i v a t i v e s o f a f u n c t i o n i n H (M) s a t i s f y a boundedness c o n d i t i o n , a Holder c o n d i t i o n and a norm i n e q u a l i t y where we might expect these p r o p e r t i e s o nly f o r the non-mixed d e r i v a t i v e s by which we define our space. -+ THEOREM 1. Let the f u n c t i o n f belong t o the c l a s s H r(M) where M^= = M R = M, and l e t the non-negative i n t e g e r s 6^ , ***» $ n s a t i s f y the i n e q u a l i t y n I SkAk < 1 ' k=l Bi + , ,* +e n 3 f Then the p a r t i a l d e r i v a t i v e s • • • • ^ e x i s t on R and belong t o the c l a s s H (M ) where P 3 x l x n r i = r i ( l " I P k / r k > ' M' = ••• = M' = M' < UlIf : L^R")!! + cM. 1 n — 1 1 P For these d e r i v a t i v e s we have the i n e q u a l i t y r : L (RN)|| < | |f : L (RN)|| + cM. In (7) and (8) the constant c does not depend f or M. -»• n 1 PROOF. Since f e H*"0M) w i t h M, = ••• = M_ = M, then A+(f)p < dM^—R P v 1 v k x k -»• s/rv f o r any v. Let v f e = 2 (s = 0, 1, * " * ) , and denote by g^ = g ggfx^,** * ,x n) an e n t i r e f u n c t i o n o f type i n the v a r i a b l e x^ which s a t i s f i e s | |f - g s : L p ( R )|| <,dnM/2s. Then the f u n c t i o n f i s represented by the s e r i e s converging i n L p ( R n ) norm s=l °0 = *0 » % = s 8 - g s _ x where ||Qg : Lp(Rn)M < 3dnM/2s (s = 1, 2, •••) A f t e r d i f f e r e n t i a t i n g t h i s s e r i e s we get g1+...+en g 1 +... +p n B 1+-«»+B n 3 f 3 Q 0 » 3 Q s Sx^ '••3x n 1 * " 3 x n n s=l 3 x x 1 * " 3 x n n where * i 1 - - - * n n p p and U8 3 1 O s 3dnM ( s ? S k / r k ) 3x, 1*«*3xT,Pn p 2 s 1 n s[( E P k / r k ) - 1 ] =(3dnM)2' 1 (s = 1, 2, • • • ) . L ine (5) i n s u r e s the v a l i d i t y o f t h i s d i f f e r e n t i a t i o n . Then the theorem i s proved by applying Lemma 2 t o the s e r i e s ( 9 ) . r We now define a norm on the space H p > and apply the previous theorem t o show completeness of the normed space. R e c a l l t h a t through-out t h i s Chapter we are usin g the HBlder c o n d i t i o n on the f i r s t d i f -ference ( f o r a < l ) o f the p - t h p a r t i a l o f f t o define the space. The norm o f the space H p(G n) w i l l be given by l | f : l£(0 n)|| = | | f : L p ( G n ) | | • M f (10) where M- i s the infimum o f constants M such t h a t f E H ((T ,M) where r P n» = ••• = M R = M; i . e . M f i s the i n f o f constants M such t h a t the f o l l o w i n g i n e q u a l i t y holds f o r a l l i : ! ! A x y x i \ h ) : L p ( G n j T i ) | | < M f h ! a i i i i f < 1, or i f a± = 1. ->• r THEOREM 2. H p i s complete. PROOF. Let { f } be a sequence of fu n c t i o n s i n H f o r which m p | | f k - f j : Hp|| * 0 as k,J * ( l l ) h9 By the previous theorem the f o l l o w i n g i n e q u a l i t i e s h o l d f o r s = 0, 1, P£.: 3 S f v a sf k - J : L ( R n ) | | < e | | f k - f , : <| |. ' (12) Thus 3x* 3x S P " " " k J x 8 S f k 8 S f i : L ( R n ) | | 0 as k„1 + », (13) axf 3xf p and from ( l l ) and (13) we conclude t h a t there e x i s t s a f u n c t i on f e L (R n) having p a r t i a l d e r i v a t i v e s f ^ (s = 0, • * *, p. ; i = 1,* * * ,n) p x^ 1 such t h a t 3 s f m 3 s f L_.(R n)! | 0 as m «. 3x* Sx* P ? Since the f u n c t i o n s f m a l l belong t o H p, each f ^ s a t i s f i e s a f i r s t or second d i f f e r e n c e c o n d i t i o n on the p x - t h d e r i v a t i v e i n x^: e.g. f o r a± < 1, i'V f*»vh > • VR n )" ± M f M a i <* - 1 . 2 » •••• »)• Thus l e t t i n g m ^ » we have the same c o n d i t i o n s a t i s f i e d f o r the -*• r f u n c t i o n f , and hence f e Rp. But from (11) and (12«) we conclude t h a t f o r m,j > N f o r N s u f f i c i e n t l y l a r g e , the q u a n t i t y i s s u f f i c i e n t l y s m a l l . Then l e t t i n g J -*•<*> ve have t h a t x x m ^ x± i s s u f f i c i e n t l y s m a l l , and we conclude t h a t -»• I I f — f : H I | -> 0 as B + » . I I m P' 1 This completes the proof. r We now t u r n t o some simple imbedding, theorems f o r the c l a s s H s r r which demonstrate the r e l a t i o n s h i p between H and W f o r i n t e g e r values o f the parameter r . p By d e f i n i t i o n , f belongs t o W i f f and i t s p a r t i a l d e r i v a t i v e s (P) n f x of order p belong t o L ( P r ) . (We of course r e f e r t o the d e f i n i t i o n 1 of f ^ as given i n Chapter 2. See Chapter 8 f o r a d i s c u s s i o n o f the equivalence t o the g e n e r a l i z e d d e r i v a t i v e . ) The norm i s given by II* : WP I! = | | f : L ( H n ) | | • || f ( p ) : L ( R n ) | | . P» xl P 1 p Note t h a t t h i s one dimensional d e f i n i t i o n corresponds t o the general case o f the Sobolev spaces as defined i n [ l U ] , but th a t the n-dim--> p r. e n s i o n a l Wp ( d e f i n e d analogously t o H p) i n v o l v e s assumptions on only the non-mixed p a r t i a l s f x ( P i ) i THEOREM 3. The f o l l o w i n g imbeddings h o l d : r r ' H •*• H , 0 < r < r ; P » x ! P,x x Note t h a t "by an imbedding ve mean not only a set t h e o r e t i c a l i n c l u s i o n , but a l s o a norm i n e q u a l i t y . Thus M + M* holds i f and only i f M c M , and there e x i s t s a constant c such t h a t f o r a l l f e M | | f : M'| J <_ c| J f : M| I. P r o p e r t i e s (ih) and ( 17) are t o be expected o f any reasonable g e n e r a l i z a t i o n o f Sobolev spaces. ( 1 5 ) stems from a l l o w i n g a = 1 i n of t h i s t heory. Thus w h i l e H p i s the wider c l a s s f o r i n t e g e r values o P+ a o f p, W i s l a r g e r than H„ f o r anv a > 0. That i s , we have the p P f o l l o w i n g t r a n s i t i v e r e l a t i o n : the end r e s u l t o f which i s already guarenteed by The proofs of p r o p e r t i e s (lU) - ( 1 7 ) are q u i t e easy and depend on Theorem k of Chapter 5 f o r the set i n c l u s i o n s and Lemma 1 o f Chapter 3 and Lemma 1 of Chapter 5 f o r the norm i n e q u a l i t i e s . As an example of the p r o o f s , we show ( 1 5 ) and ( 1 6 ) . PROOF OF ( 1 5 ) . Let f e Wp . Then f , t^t) e L ( F n ) and bv Lemma 1 of p,x^ j. p Chapter 5 , the d e f i n i t i o n of H* The r e l a t i o n ( l 6 ) then shows the completeness P Thus f e l P . * l and furthermore 52 and we conclude ( 1 5 ) . PROOF OF (16). Let f e H where r = p + a. Since 0 < p, x , ~~ f v. e L_(R ) and w e conclude t h a t f e Wp . Now bv Lemma 1 of x l p P,x x Chapter 3, ! Lp(Rn)" i c l H f 8 V^^! + C 2 M f whence Mf : WS',x II id+^JMf : L p ( R n ) | | + c 2 M f < c | | f : H j t X i | | . This completes the proof o f ( l 6 ) . CHAPTER SEVEN AN IMBEDDING THEOREM FOR THE SPACE H p While Theorem 3 o f the previous Chapter showed r e l a t i o n s h i p s r P P among the c l a s s e s H _ , H , and W , i t considered only spaces P.*!' P,x x' p,x x» v d e f i n e d over a l l o f R n having the same value o f p. The f o l l o w i n g theorem, the r e a l culmination o f t h i s work, gives c o n d i t i o n s on the -+-parameters r x , ***, r n f o r the imbedding o f H p i n t o s i m i l a r spaces defined over lower dimensional subspaces of R* and having d i f f e r e n t values o f p. I f we put r ± = *** = r R = r , t h i s theorem i s a c t u a l l y a g e n e r a l i z a t i o n o f the Sobolev imbedding theorem f o r Wp. (See [2] and [15].) THEOREM 1. Suppose 1 <_ p <_ p < «° and > 0 , i * ,1, n. For 1 < m ««n l e t k = 1 - l / p ( I l / r ± ) + l / p ' ( I l / r ± ) > 0. m i = l i = l •f + n I f f ( x 1 , * " , x n ) belongs t o the c l a s s H p(M) on R , then f o r f i x e d x m + x , * * " , x n the f u n c t i o n f considered as a f u n c t i o n o f the v a r i a b l e s xl****»xHl belongs t o the c l a s s H ,(M ) on i f where p ,5=1 J w i t h the constant y depending only on n, and 6 depending only on n, m, r ^ , •**, r n # Thus we have the imbedding Hp(B.n) * H^R* 1) . PROOF. Let f e Hp(M) and l e t v ^ 1 * 2 S f o r s = 0, 1, '' *. Then by Theorem 2 of Chapter 5, ^ W p i (*I V / 2 ' • ^ k=l Since - v^(s) i s a f u n c t i o n o f s, denote by g g an e n t i r e f u n c t i o n of type i n the v a r i a b l e x k s a t i s f y i n g | | f - g : Lp(R n)|[ < (d I M k)/2 S . k=l Then the f u n c t i o n f can be represented by the s e r i e s (which converges i n the L ( R n ) norm): OO 00 f = g 0 + I ( g s - g 8_!) = Q 0 + I % . s=l s=l Then ||Q : L_(R n)|| <^  (2d J M . ) ^ 8 " 1 (s = 1, 2, k=l S / rir and Q s i s ah e n t i r e f u n c t i o n o f type 2 i n the v a r i a b l e x^. Consequently, by l i n e s (10) and (12) of Chapter k f o r the subspace R m of R n of p o i n t s u x , * ** .u^ . x ^ , • • • ,x n f o r f i x e d x ^ , ' " . ^ the norm of Q s i n R m s a t i s f i e s ||Q8 : Lp.tR*)!! = = (J •••/ | Q g ( u 1 . " - , u m , x n + 1 , - - - , x n ) | P d u 1 — d u f f l ) 1 / p ' 56 <_2n ( n v K ) 1 / P , | | Q S : L .<F»)|| m+1 * 1 m+1 p < 2 2 n + 2 ( d K ) / 2 8 \ k=l k Therefore s=l s=y < 2 2 n + 2 d K J - i — < (c l M k ) / 2 P K m k-1 k s=u 2 S ^ ~ k-1 k vhere the constant c depends on r and < m. Since QQ + £ Q- i s o f type 1 8 u AM no g r e a t e r than 2 i n the v a r i a b l e x x , s=l p k=l 1 By Theorem 3 of Chapter 5, t h i s means t h a t f - g 0 as a f u n c t i o n of Xi »** * ,x. belongs to the c l a s s H 8^" 1^(R m,Mj) where n w i t h C g / \ defined as i n l i n e (22) o f Chapter 5 (a = 2 i f r = s.(m)), Consequently, f - g Q e ^ '(M) f o r any f i x e d x m + 1 , # * * , x n . Now the f u n c t i o n g i s o f type u n i t y i n a l l the v a r i a b l e s x^^, and from i n e q u a l i t y (3) ll«o 5 V R n ) H 1 H f 5 V R n ) l l + d I M k • and by l i n e s (10) and (12) of Chapter h we have th a t | | g 0 : L p,(P » )H < 2 n ! | g 0 : L p , ( K n ) | | < 2 2 n | | g 0 : 1^)11 < 2 2 n ( | | f : L (R*1) 11 + d f ^ ) . p k=l Thus by Theorem h of Chapter 5 i t f o l l o w s t h a t g Q belongs t o the c l a s s where But the sum of the constants and cannot exceed n Y | | f : L p ( * n ) ( | + B I M k , k=l which shows th a t f e H p»(M ) f o r any f i x e d x m + x » * * * » x n where M* < Y | | f : l^mW + B [ M k , and t h i s completes the proof. We remark tkat this transformation f*om r x t o s^m) v i a tc m i s t r a n s i t i v e . Indeed, i f we have numbers r x , • " , r n and p w i t h > 0 i ii t tt and 1 <_ p <_ 0 0 and then chose numbers p and p w i t h p <_ p <_ p and spaces R" and R J w i t h j < m < n, then the transformation from (p,R ) to (p'jR 1 1 1) and then from (p'jR 0 1) t o (p",R^) r e s u l t s i n parameters SX(<1)» **** snCj) which can be obtained by going d i r e c t l y from (,p,Rn) t o (p",R- 1). 58 Nov i t i s p o s s i b l e t o demonstrate the f o l l o w i n g r e s u l t which shows that the c o n d i t i o n ( l ) may not be weakened. I f 1 - l / p ( ^ 1 /^) > 0, 1 < p, and e > 0, then f o r any p' and m w i t h 1 < p _< p' and 1 <^  m f _ n , there e x i s t s a f u n c t i o n f belonging t o H** but not ( f o r any constant M) t o H_»(M) where s = (s 1(m),* ,*,s i_ 1(m),s i(m)+e,s i + 1(m) f"«,s i n(m)). In E'9] pp. 2T0-27H ( o r pp. 30-33 i n the t r a n s l a t i o n ) N i k o l ' s k i i gives an example of a f u n c t i o n s a t i s f y i n g the above statement f o r the case i m = 1 and p = ». But by the remark on t r a n s i t i v i t y , t h i s t r a n s f e r from (p,RnH.to (»,R) s u f f i e i e s t o cover a l l the cases (p* .R™) w i t h p < p', and 1 < m < n. Since c o n d i t i o n ( l ) cannot be weakened, Theorem 1 i s a complete and f i n a l r e s u l t f o r the H-spaces. This i s r a t h e r unexpected because the c o n s t r a i n t t h a t <m > 0 would appear t o be der i v e d s o l e l y from the t e c h n i c a l i t i e s i n the proof f o r convergence of the s e r i e s (.5) of e n t i r e f u n c t i o n s . This apparently demonstrates the p r e c i s i o n of the approximation by e n t i r e f u n c t i o n s . At the outset i t was mentioned t h a t these r e s u l t s are d i f f e r e n t i n form from those o f Sobolev f o r i n t e g r a l parameters r . But they are i n f a c t g e n e r a l i z a t i o n s o f some o f Sobolev's theorems i n the sense t h a t the non«*mixed p a r t i a l s d efined here are a c t u a l l y the g e n e r a l i z e d der-i v a t i v e s . Suppose the f u n c t i o n f ( x 1 , , , * , x _ ) has the non-mixed p a r t i a l f ^ ( 1) or order P (more p r e c i s e l y , f has a d e r i v a t i v e f P a b s o l u t e l y X l continuous on any f i n i t e i n t e r v a l r e l a t i v e t o x^ f o r almost a l l x2» *n) w h i c h "belongs t o L p ( R n ) f o r almost a l l x 2,*"»x n. I f <P i s any f u n c t i o n which, together w i t h i t s p a r t i a l d e r i v a t i v e s up tc order p, i s continuous snd v a n i s h i n g on the boundary of the cube Qc c o n s i s t i n g of p o i n t s ( x 1 , * " , x n ) which s a t i s f y — x ± f.bi» t h e n a f t e r p - f o l d i n t e g r a t i o n by p a r t s we get the equation b l 9 p f o b l 3 ° f / f p~ d x i - (-D J * —T a-^  dx^ B.^ 3x^ which holds f o r almost a l l x 2» *"» XJJ. A f t e r i n t e g r a t i n g w i t h r e -spect t o ( x g * * * * , ^ ) we have »P* p+1 j ( f -+ (-1) «P*)dx = 0, fl 3 x x p 3 p f and we conclude t h a t * = i s indeed the ge n e r a l i z e d d e r i v a t i v e . 3 x l C onversely, l e t $ e Lp(.R ) be the g e n e r a l i z e d d e r i v a t i v e of f i n the v a r i a b l e x-^. As shown by Sobolev ([lU], s e c t i o n 5) there corresponds t o the f u n c t i o n f an averaging f u n c t i o n ( " m o l l i f i e r " ) f u e L (R n) depending on h which has continuous p a r t i a l d e r i v a t i v e s of any order such t h a t | | f - f h : L p ( R n ) | | •* 0 as h •+ « " 3 \ n and p * * n e Lp(,R ) where 3 x l 3 p f *l! - : L p ( R n ) | | + 0 as h + 3x^ But by Lemma 2 of Chapter j> we then conclude t h a t f may be changed on a set of measure zero so t h a t 3 p f 3X-,0 Now i t i s e a s i l y seen t h a t Theorem 1 i s a g e n r e a l i z a t i o n , when r ^ = ••• s r n = r and f o r spaces defined over R n, of some theorems of Sobolev. In f a c t , l e t the f u n c t i o n f e L (R n) have a l l g e n e r a l i z e d d e r i v a t i v e s o f order r ( r an i n t e g e r i n t h i s instance) which belong t o L p ( R n ) . Then the non-*nixed p a r t i a l d e r i v a t i v e s are the usua l der-i v a t i v e s d efined h e r e , and by Lemma 1 of Chapter 5» ||AX ( f ( x " X ) , h ) : L ( R n ) | | x l x l p i h | | A X i ( f ( ^ . h ) : LpCP")!! < 2 h | | f ( x > : L p ( R n ) | | 1 and thus f belongs t o the c l a s s H (M) where r = ( r , "' ,r) and M = ( , • • • ) w i t h M± = 2 | | f ^ J ^ : L p ( R n ) | | . Consequently, the fu n c t i o n f , considered on the l i o e a r subspace x^, x ^ belongs t o Hp«'(M*) where s • (s,***,s) w i t h s = r i c m = r - m/pf + n/p > 0, and <_ Y ! \? : L p ( R n ) | | + SnMj^ , ( l < _ i < t i ) . Thus i f we l e t s = p + a, p an i n t e g e r and 0 < a < 1, then the f u n c t i o n f has p a r t i a l d e r i v a t i v e s o f order p w i t h respect t o each x j which belong t o L p i f R 8 1 ) . We may obta i n estimates f o r the Lp^R 1 1 1) norms of these d e r i v a t i v e s by e s t i m a t i n g the s e r i e s (h) u s i n g (5) termwise. 6 l Furthermore i f r ± = *** = r R = r i n Theorem 1, we have only the s i n g l e c o n d i t i o n t h a t n m Km = 1 + — > 0 pr p r or e q u i v a l e n t l y , i mp P < • n-rp I t now f o l l o w s immediately t h a t the non-mixed p a r t i a l s o f f e x i s t and belong t o Lpi^R" 1). For a s i m i l a r r e s u l t f o r the mixed p a r t i a l d e r i v a t i v e s we r e c a l l Theorem 1 of Chapter 6. CHAPTER EIGHT THE CLASSES Hp*(G n), Wp*(Gn), Bp*(Gn) In t h i s f i n a l Chapter ve make b r i e f comment on the e n t i r e spectrum of f r a c t i o n a l order Sobolev spaces, and note t h e i r r e l a t i o n s h i p s w i t h the spaces already mentioned and the p r o p e r t i e s thereof. We again assume t h a t 1 < p < « and t h a t G i s an open subset of — — n R n. Then f o r non-negative i n t e g e r s p, f = f ( x x »* * * ,x n), belongs t o the c l a s s Wp(G n) i f the f o l l o w i n g norm i s f i n i t e : I I * « W p(G n)|| - [||f : V G n M I P • El|f t p > t V G n ) M P l 1 / P where the sum i s taken over a l l g e n e r a l i z e d p a r t i a l d e r i v a t i v e s o f order p. Now f o r any r > 0 we examine the f r a c t i o n a l order W, B, and H spaces which provide two b a s i c c l a s s i f i c a t i o n s f o r the continuous f u n c t i o n s . The f i r s t i s the f a m i l y of clas s e s W5(G ) which i s n a t u r a l l y completed by the c l a s s e s B p ( G n ) . The second c l a s s i f i c a t i o n i s made up of the cl a s s e s Hp(G n) as already d e f i n e d , where we make the convention -V t h a t i f r ± = •** = r n = r , then we w i l l w r i t e Hp"(Gn) f o r H*(.Gn). L e t t i n g r = p + a as b e f o r e , i f o 4 1, then f belongs t o Wp(G n) i f the f o l l o w i n g norm i s f i n i t e : l|f * WJCGJM = f | |f : yo n>|| P* U( P ) , x Jp), .,P jf ( x ) - f ( y ) | x / + ^ i — ; ^ ** & 1 ° n Gh l x - yl where the sum i s taken over a l l g e n e r a l i z e d p a r t i a l .derivatives of order P y I f a * 1, then f belongs t o w p ( G n ) i f the norm ( l ) i s f i n i t e . To define Bp(G Q) t i f a 4 1, then Bp(C»n) = W J J C O j j ) . and i f a = 1, f belongs t o B p ( G n ) i f the f o l l o w i n g norm i s f i n i t e : | | f : B j ( G n ) | | = [ | | f : Lp(Gn>MP + | f ( p ) ( x ) - 2 f ( P ) ( ( x + y ) / 2 ) • f ( p ) ( y ) | P l / p + IJ dxj • — dy] G n Gn,x I* - yl where G n x i s the set of a l l p o i n t s y e G n such that (x + y ) / 2 c G n, and where the sum i s again taken over a l l g e n e r a l i z e d p a r t i a l der-i v a t i v e s of order p. The c l a s s H r(G ) i s defined as i n Chapter 2 w i t h norm as given i n p n Chapter 6. Henceforth we assume t h a t 1 < m < n and G i s a s u f f i c i e n t l y — — m smooth m-dimensional submanifold o f G n. We now s t a t e the w e l l known general imbedding theorem f o r ^ ( C ^ ) w i t h i n t e g r a l p. THEOREM 1. I f p i s an i n t e g e r and 0 < r = p - n/p + m/p , 1 < p < p < », then t W P(G n) - W P t(G m) p n p m i i where p i s the i n t e g r a l p a r t o f r . R e c a l l t h a t any imbedding e n t a i l s a norm i n e q u a l i t y ( c f . Chapter i To consider the c l a s s Hp(G n), we have already proved Theorem 1 of Chapter 7 f o r R n which s t a t e s f o r G th a t r ' = r - n/p + m/p* > 0 , 1 < p' to« Now r e c a l l the comment at the end of Chapter 6 which r e f e r s t o the f o l l o w i n g r e l a t i o n s h i p f o r any e > 0 : p n p n p n which holds f o r p an i n t e g e r . And at the end of the proof o f Theorem 1 of Chapter 7 we s t a t e d the t r a n s i t i v e nature of the imbedding ( 3 ) . Two consectutive t r a n s i t i o n s from (p,n) t o (p',m) and then from (p',m) t o (p , j ) may be replaced by one d i r e c t t r a n s i t i o n from (p,n) t o (p In these senses, t h e n , the f a m i l y of cl a s s e s H^(G ) i s c l o s e d w i t h P n respect t o imbeddings. P r We ask i f the c l a s s e s W p ( G R ) and t h e i r g e n e r a l i z a t i o n s Wp(G n) posses such a c l o s e d property w i t h respect t o imbedding theorems. Un-f o r t u n a t e l y they do n o t , as i n d i c a t e d by the f o l l o w i n g theorem. THEOREM 2 . I f 1 < p < p' <_» and r ' = r - n/p + m/p', then the imbedding holds only under the f o l l o w i n g c o n d i t i o n s : i ) p < p and r ' > 0 ; i i ) p = p' 4 2 w i t h r* > 0 and r ' n o n - i n t e g r a l ; i i i ) i f p = p' = 2 , then we have the f o l l o w i n g imbeddings without e x c e p t i o n , t • W£(Gn) Wg (G m)., i f r ' > 0 . r r R e c a l l t h a t f o r r > 0, r n o n - i n t e g r a l , S p ( ^ n ) = ^ p ^ G n ^ # W e r r a l s o have t h a t f o r a l l r , B 2 ( 0 n ) = W 2 ^ G n ^ * 7 1 x 1 1 3 t h e ^ P o r t 8 1 1 0 6 ° f the B c l a s s e s , which d i f f e r from the W c l a s s e s only f o r i n t e g r a l r , i s shown by the f o l l o w i n g two theorems, the f i r s t o f which shows t h a t the B c l a s s e s posses a closed system o f imbeddings as do the H c l a s s e s . THEOREM 3. I f 1 <, p <_ p' <,00 and r ' = r - n/p + m/p' > 0, then *p(cn) * B J : ( O B ) . F i n a l l y we f i n d t h a t the completeness o f the W c l a s s e s w i t h respect t o imbeddings can be achieved only by the i n f u s i o n o f c e r t a i n B - c l a s s e s . THEOREM k. I f 1 < p < p <_«> and r = r - n/p + m/p > 0, then WJ(0 ) - B * \ ( G ) . F n p m Jbr f u r t h e r information on the H, B, W cl a s s e s f o r any r > 0 we c i t e the f o l l o w i n g references. For the use of f r a c t i o n a l L i o u v i l l e d e r i v a t i v e s i n d e f i n i n g the W c l a s s e s and an i n t e r e s t i n g geometrical treatment o f imbedding theorems, see Sobolev and N i k o l ' s k i i , [ l 6 ] , s e c t i o n 1. For an a n a l y t i c d e f i n i t i o n o f the f r a c t i o n a l d e r i v a t i v e s and t h e i r use i n the d e f i n i t i o n of W c l a s s e s , see K i p r i y a n o v , [h] and [ 5 l . r N i k o l ' s k i i i n [ l l ] s e c t i o n 2 gives a survey of r e s u l t s on H and P »xi P i W , wh i l e s e c t i o n s 3 and h give a thorough resume of p r o p e r t i e s o f P , X J 7 r t H p ( G n ) , B p ( G n ) and W p(G n). M n a l l y , S l o b o d e t s k i i has pub l i s h e d many works on the W c l a s s e s as def i n e d i n t h i s s e c t i o n . We r e f e r t o [13]. 66 BIBLIOGRAPHY 1. Ah k i e z e r , N. I . Lectures i n the Theory of Approximations, Nauk, Moscov. 1965T (Russian). 2. C l a r k , C, W. I n t r o d u c t i o n t o Sobolev Spaces, Seminar Notes, U n i -v e r s i t y o f B r i t i s h Columbia, Vancouver. 1968. 3. Hardy, G. H., J . E. L i t t l e w o o d , and G. P<51ya. I n e q u a l i t i e s , Cambridge U n i v e r s i t y P r e s s , London. 193k. k. K i p r i j a n o v , I . A. The F r a c t i o n a l D e r i v a t i v e and Imbedding Theorems, Dokl. Akad. Nauk SSSR, 126 (1959h 1187-1190. (Russian). 5. K i p r i j a n o v , I . A. On Spaces o f F r a c t i o n a l l y D i f f e r e n t i a b l e Fun-c t i o n s , I z v . Akad. Nauk SSSR Ser. Mat. 2k (i960), 865-882. (Russian). 6. N i k o l ' s k i i , S. M. Extension of Functions of Seve r a l V a r i a b l e s P r e s e r v i n g D i f f e r e n t i a b l e P r o p e r t i e s , Mat. Sb. kO (82) (1956); E n g l i s h T r a n s l . AMS T r a n s l . (2) 83 (1969), 159-188. 7. N i k o l ' s k i i , S. M. The G e n e r a l i z a t i o n o f One P r o p o s i t i o n o f S. N. Bernshtein o f D i f f e r e n t i a b l e Functions o f Se v e r a l V a r i a b l e s , Dokl. Akad. Nauk SSSR, 59 (9) il9kQ) t 1533-1536. ( R u s s i a n ) . 8. N i k o l ' s k i i , S« M. Imbedding Theorems f o r Functions v i t h P a r t i a l D e r i v a t i v e s Considered i n Various M e t r i c s . I z v . Akad. Nauk SSSR Ser. Mat. 22 (1958), 321-336; E n g l i s h T r a n s l . AMS T r a n s l . (2) 88 (1970), 1-U0. 9. N i k o l ' s k i i , S, M. I n e q u a l i t i e s f o r E n t i r e : f u n c t i o n s o f Exp o n e n t i a l Type and T h e i r A p p l i c a t i o n t o the Theory of D i f f e r e n t i a b l e Functions o f Se v e r a l V a r i a b l e s , Trudy Mat. I n s t . S t e k l o v , 38 (1951), 2^-278; E n g l i s h T r a n s l . AMS T r a n s l . (2) 80 (l96l), 1-38. 10. N i k o l ' s k i i , S. M. Of One Property o f the Class H_*, Ann. t h i v . S c i . Budapest, Eotvos Sect. Mat. 3-k (1960/1961), 205-216. (Russian). 11. N i k o l ' s k i i , S. M. On Imbedding, C o n t i n u a t i o n , and Approximation Theories f o r D i f f e r e n t i a b l e F i i n c t i o n s o f Se v e r a l V a r i a b l e s , Russian Math Surveys, l6 (5) 1961, 55-10U. 12. N i k o l ' s k i i , S. M. P r o p e r t i e s o f C e r t a i n Classes o f Functions of Se v e r a l V a r i a b l e s on D i f f e r e n t i a b l e M a n i f o l d s , Mat. Sb. 33 (75) (1953), 261-326; E n g l i s h T r a n s l . AMS T r a n s l . (2) 80 (1969), 39-118. 67 13. S l o b o d e t s k i i , L. N. On Sobolev's Space o f F r a c t i o n a l Order and i t s A p p l i c a t i o n t o Regional Problems f o r P a r t i a l D i f f e r e n t i a b l e Equations, Dokl. Akad. Nauk SSSR, 118 (2) (1958), 2U3-2U6. (Russian). lh. Sobolev, S. L. A p p l i c a t i o n s o f F u n c t i o n a l A n a l y s i s i n Mathematical  P h y s i c s . American Mathematical S o c i e t y , Providence, R.I. 1963. 15. Sobolev, S. ,L. Of One Theorem o f F u n c t i o n a l A n a l y s i s , Mat. Sb. k (U6) (1938), U71-VT9. (Russian). 16. Sobolev, S. L. , and S. M. N i k o l ' s k i i , Imbedding Theorems, Proc. Fourth A i l - U n i o n Math. Congress (Leningrad, 196l), I z d a t . Akad. Nauk SSSR, Leningrad, I963. 227-2*42; E n g l i s h T r a n s l . AMS T r a n s l . (2) 87 (1970), 1U7-173. 17. Y o s i d a , Kosaku. F u n c t i o n a l A n a l y s i s . S p r i n g e r - V e r l a g , B e r l i n . I96U. 18. Zygmund, A. Trigonometric S e r i e s , V o l . I I . Cambridge U n i v e r s i t y P r e s s , London. 1959. 

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