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Interpolation theory and Lipschitz classes on totally disconnected groups Bradley, John Scott 1974

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INTERPOLATION THEORY AND LIPSCHITZ CLASSES ON TOTALLY DISCONNECTED GROUPS by JOHN SCOTT BRADLEY B.Sc, McMaster University, 1973 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1974 In p resent ing t h i s t h e s i s in 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 fo r reference and study. I f u r t h e r agree tha t permiss ion for e x t e n s i v e copying o f t h i s t h e s i s fo 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 . It 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 a l lowed without my w r i t t e n p e r m i s s i o n . 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 - i i -ABSTRACT This thesis concerns the absolute convergence of the Fourier series of functions belonging to certain Lipschitz classes on totally disconnected groups. The technique used i s one of interpolating between certain endpoint results which are proven directly. These results are shown to be best possible and a counterexample i n interpolation theory i s given. - i i i -TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER ONE INTERPOLATION THEORY 2 CHAPTER TWO LIPSCHITZ CLASSES ON TOTALLY DISCONNECTED GROUPS 17 CHAPTER THREE APPLICATIONS OF INTERPOLATION THEORY 29 BIBLIOGRAPHY 43 - i v -ACKNOWLEDGEMENTS I wish to express my appreciation to my thesis supervisor, Dr. J . Fournier, for h i s assistance and encouragement during the w r i t i n g of t h i s t h e s i s . I also wish to thank Dr. J . Coury f o r h i s help and assistance i n reading t h i s work and Mrs. Y i t - S i n Choo for typing i t . The f i n a n c i a l assistance of the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council i s also acknowledged. - 1 -INTRODUCTION This thesis i s composed of three chapters. The f i r s t i s a discussion of some r e s u l t s i n i n t e r p o l a t i o n theory. Due to the length and complexity of the proofs involved they have been omitted. The reader i s refered to [3] and [9] f o r d e t a i l s . A b r i e f d i s c u s s i o n of V i l e n k i n groups i s to be found at the beginning of chapter two. The remainder of the chapter concerns r e l a t i o n -ships between the smoothness of functions and the s i z e of t h e i r Fourier c o e f f i c i e n t s . Extensions of the r e s u l t s of chapter two, together with a b r i e f d i s c u s s i o n of the sharpness of our r e s u l t s , form the bulk of chapter three. i A counterexample i n i n t e r p o l a t i o n theory i s also given. The technique used i n chapter three to extend the r e s u l t s of chapter two i s that of i n t e r p o l a t i n g between endpoint r e s u l t s discussed i n chapter two. We make no claim that t h i s i s the only way to achieve these r e s u l t s ; however, i t does provide a u n i f i e d treatment of what would otherwise seem to be an unrelated c o l l e c t i o n of c r i t e r i a . I t i s suggested that the reader who i s unfamiliar with i n t e r p o l a -t i o n theory should begin by reading chapters two and three, r e f e r r i n g back to one when necessary. Theorems 2.9 and 2.10 together with a l l of chapter three are new. A l l other r e s u l t s can be found i n the l i t e r a t u r e . - 2 -CHAPTER ONE INTERPOLATION THEORY We begin our discussion of i n t e r p o l a t i o n theory by quoting the theorems of Riesz-Thorin and Marcinkiewicz. The concepts involved are then formally generalized and extended to l a r g e r classes of spaces and maps. To begin, we need the following d e f i n i t i o n s . D e f i n i t i o n 1.1 : Let L P(X, u) and L q ( Y , v) be the usual Lebesgue spaces. I f Y i s a l i n e a r operator from iF to L q we say T i s of (strong) type (p, q) i f | | T f | | q _.M|.|f|| f o r a l l f £ L P and we define the (p, q) norm of T to be the infimum of a l l such M's. Then Theorem 1.2 : (Riesz-Thorin) Let T be of type ( p Q , q Q) and ( p ^ q ^ with norms M q and r e s p e c t i v e l y . Let 0 £_ 0 _< 1 and define — = + — , — = + — . Then T i s of type (p, q) and the (p, q) P P 0 p l q q o q l 1—0 0 norm of T i s at most M M, . o 1 However powerful we might f e e l t h i s theorem to be we s h a l l see that somewhat weaker conditions on the "endpoints" (p^, q^), i = 0, 1 , often s u f f i c e to insure strong type (p, q) for the "intermediate spaces". We introduce the following concepts. - 3 -D e f i n i t i o n 1.3 : Let f be a measurable function. The d i s t r i b u t i o n function of f i s defined by D f(x) = y{t : |fCt)| > x> . I f T i s an operator from some l i n e a r space of measurable func-tions to another space of measurable functions, T i s subadditive i f f o r almost a l l values of x we have I T C ^ + f 2 ) (x) | <_ J T f ^ x ) ! + | T f 2 ( x ) | . D e f i n i t i o n 1.4 : Let the domain of T contain a l l f i n i t e l i n e a r combina-tions of the c h a r a c t e r i s t i c functions of sets of f i n i t e measure and a l l truncations of a l l i t s members. When 1 <_ p <_ °° and 1 <^  q < 0 0 we say that T i s of weak type (p, q) i f there e x i s t s a constant c u i t h e ! I f ! i . D T f ( y ) < In the case where q = 0 0 we define weak type to be the same as strong type. We define the weak (p, q) norm of T to be the infimum of a l l c's s a t i s f y i n g the above i n e q u a l i t y . It follows r e a d i l y that strong type (p, q) implies weak type (p, q) but not conversely (see f o r example [10]). We can now state the Marcinkiewicz i n t e r p o l a t i o n theorem. Theorem 1.5 : Let T be a subadditive operator of weak type ( p Q , q^) and (p^, q^). Furthermore suppose 1 p^ <^  q^ <_ 0 0 f o r i = 0, 1 and q Q £ q^. - 4 -I f 0 < 9 < 1 and - = — + — , - = — + — , then T i s of strong P Pc P l q q Q q x type (p, q) . Dir e c t proof of theorems 1.2 and 1.5 can be found i n [9]. The immediate question would seem to be whether the concepts of these two theorems can be extended i n some sense to other l a r g e r c l a s s e s of spaces. The answer i s yes. To see t h i s we begin with the following concepts. Let A be a l i n e a r Hausdorff space and l e t A q and A ^ be two Banach subspaces of A such that the i n j e c t i o n s of A ^ ( i = 0, 1) are continuous. We c a l l such a p a i r ( A Q , A ^ ) an i n t e r p o l a t i o n p a i r and define the i n t e r s e c t i o n A O A , and the algebraic sum A + A . i n the normal o 1 & o 1 manner. Much of the following material p a r a l l e l s the discussions i n Butzer and Berens [3] and Stein and Weiss [ 9 ] . The reader i s r e f e r r e d to these works for any of the proofs which are omitted. Theorem 1.6 : The spaces A Q Pl A ^ and A Q + A ^ are Banach spaces under the norms f N A nA. = M A X o 1 £ I I A - I I ' N A , o 1 I I ^ A + A = /f+f HfoMA + ' M A . o 1 f=f +f. ^ o 1 o 1 f i e A i Furthermore A Q n A ( ) + A 1 (i = 0, 1). - 5 -D e f i n i t i o n 1.7 : An intermediate space (of A q and A^) i s a Banach space A C A such that A n A, C A C A + A. . o 1 o 1 The obvious example of an intermediate space i s seen i n theorems 1.2 and 1.5. I f and are Lebesgue spaces, then the space L P , where — = + — f o r 0 < 8 < 1, are intermediate between L ° and L^^". P P G P l - -Later we w i l l see that a more general c l a s s of spaces, namely the Lorentz spaces, are intermediate between the Lebesgue spaces. There are several methods of generating intermediate spaces. For the purposes of t h i s t h e s i s we w i l l discuss only two of the s o - c a l l e d " r e a l v a r i a b l e " methods; the K-method and the J-method. We w i l l not discuss the "complex-variable" method used by Calderdn i n [ 4 ] . We make the following d e f i n i t i o n s f o r 0 < t < 0 0 . D e f i n i t i o n 1.8 : For f e A n A. set o 1 J ( t , f) = max (||f|| A , t | | f | | A ) o 1 D e f i n i t i o n 1.9 : For f e A Q + A ^ set K (t , f) = inf (||f o || A + t U f J I ) f=f +f-o 1 D e f i n i t i o n 1.10 : For any Banach space X we define L q(X) to be the Banach space of a l l classes of functions t — > g ( t ) , where g(t) e X and dt the map t — > g(t) i s strongly measurable with respect to the measure — - 6 -(Haar measure with respect to the m u l t i p l i c a t i v e group of points t e ( 0 , ° ° ) ) , and the norm U(-) | | L q ( x ) -r°° - ^ 1 /q | | g ( t ) | | q ^ . 1 < q < - , 0 ' ess sup | |g(*) | | , q = 0 0 , i s f i n i t e . D e f i n i t i o n 1 . 1 1 : The space ( A , A , ) . „ i s defined to be the space of r o' 1 9,q;K a l l f e A + A . such that t ~ 9 K ( t , f) e L ? . o 1 * We w i l l only be i n t e r e s t e d i n the cases 0 < 8 < 1 , 1 <^  q < °° and/or 0 •<_ 8 <_ 1 , q = 0 0 as i n a l l other cases the spaces contain only the zero element. Proposition 1 . 1 2 : The spaces ( A Q , q q # R for 0 < 8 < 1 , 1 £ q < » and/or 0 8 <_ 1 , q = °° are Banach spaces under the norm L i , Furthermore A 0 A . C , ( A , A , ) „ „ CL A + A , . p 1 v o' l ' S . q j K o 1 Thus we observe that i n the n o n t r i v i a l cases the spaces ( A , A , ) . „ are intermediate spaces of A and A , . I t turns out that o' 1 8,q;K K o 1 A <LZ ( A , A^)n T r and A , <LZ ( A , A , ) . T , . o o' rO,»;K 1 o' I ' I J O O J K - 7 -D e f i n i t i o n 1.13 : For - 0 0 < 8 < °° and 1 £ q £ 0 0 we define ( A , A , ) „ T to be the space of a l l f £ A + A , such that there e x i s t s v o' 1 6,q;J o 1 a strongly measurable (with respect to ^ ) function u : (0, °°) -> A Q + A ^ such that f = u(t) ^ , (u(») e L2(AQ + A X ) ) , t ~ G J ( t , u ( t ) ) e . Theorem 1.14 : The spaces (A , A..) are meaningful f o r 0 <_ 8 <_ 1, O J - tj J q = 1 and/or 0 < 8 < 1, 1 < q j< °° . Under the norms llflL a . j = i n f ( | | t - 6 J ( t , u ( t ) ) | | ) r dt L * f= u ( t ) ^ -they are Banach spaces with A H A, C (A^, A 1) C A + A. . J o l o 1 8,q;J o l Thus we see that we have again generated an intermediate space of A q and A^. An alt e r n a t e c r i t e r i o n f o r f e A Q + A^ to belong to t h i s intermediate space can be shown. Theorem 1.15 : A function f belongs to ( A , A ) f o r 0 < 8 < 1, o j. o,q,j q = 1 and/or 0 < 6 < 1> 1 < q < « i f f there e x i s t s an i n f i n i t e l y often, strongly, d i f f e r e n t i a b l e function u = u ( t ) , 0 < t < « i n A q ft A ^ such that , 0 0 f = u(t) ^ - , (u(-) e L*(A n A )) Q t o 1 — 6 Q and t J ( t , u ( t ) ) e - 8 -In the course of proving t h i s one sees Theorem 1.16 : A H A , i s a dense subspace of (A , A.). T f o r o 1 o 1 H,q;J 1 < q < 0 i f 0 < 9 < 1, and for q = l i f 9 = 0 or 1. Under c e r t a i n conditions the K- and J - methods are equivalent. In p a r t i c u l a r we have Theorem 1.17 : For 0 < 0 < 1, 1 < q < «, (A , A ) = (A , A ) — — o 1 9,q;J o 1 b,q;K. with equivalent norms. Also and ( V Vo,l;J c A o C <V Vo.-ji ( V V 1 . U J C A l C ( V V l , - ; -Further f o r 1 .£ p £ q £ °°, 0 < 8 < 1, we have ( V V6,P;K ^ <V A l > 9 , q ; K " In p a r t i c u l a r f o r 0 < 6 < 1, l<^q_<°°, (A , A..). T, C Z (A , A,) r t o' 16,q;K ^- v o' 1 9,«;K We also have K(t, f) l c t 6 | ! f | | 8 ) q ; K , - 9 -where c i s a constant depending only on 0 and q. We now wish to discuss the theory of r e i t e r a t i o n but f i r s t we need the following d e f i n i t i o n s ( A w i l l be an intermediate space of A Q and A ^ ) . D e f i n i t i o n 1.18 : A belongs to the c l a s s K(0; A Q , A ^ f o r 0 £ 0 <_ 1 i f f there e x i s t s a constant c^ with K(t, f) <_c^«t 9||f||^ f or any f e A. D e f i n i t i o n 1.19 : A belongs to the c l a s s J ( 0 ; A q , A ^ f o r 0 <_ 0 <_ 1 —0 i f f there e x i s t s a constant such that II^IIA — ° 2 t ^^t' ^  ^ o r f E A O A . , o 1 D e f i n i t i o n 1.20 : A belongs to the c l a s s H(6; A q , A 1 ) , 0 <_ Q <_ 1 i f f A belongs to both K(9; A q , A 1) and J ( 0 ; A q , A^) . These classes can be characterized i n the following manner. Theorem 1.21 : An intermediate space A of A q and A^ belongs to (a) K(05 A , A ), 0 < 0 < 1 i f f A C (A , A ) . o ± — — o i 9,«>;K (b) J(e; A Q, A ^ , 0 < 6 < 1 i f f (A Q, C_ A . (c) H(Q; V V ' 0 - 9 - 1 l f f (A , A-) . CI A C (A , A 1) v . o 1 e , l ; J o' rQ,«>;K - 10 -As an immediate c o r o l l a r y we see Cor o l l a r y 1.21.1 : For 0 < 9 < 1, 1 £ q £ °°, ( A q , A ^ e H(9; A q , A ^ and A e H(0; A , A,), A, e H ( l , A , A,), o ' o' 1 ' 1 o 1 We are now ready to state the theorem of r e i t e r a t i o n . Theorem 1.22 : Let A. and A- be two intermediate spaces of A and e 1 9 2 A± belonging to H(8 ; A q , A ^ and H(9 2; A o > A ^ , (0 1 8 ^ 8 2 < 1) re s p e c t i v e l y . Write 9 = (1 - 9 , ) 9 1 + 8'9 2 , (0 < 9' < 1) and l e t 1 < q < - . Then ( A ^ , A ^ , ^ , = ( A Q , \ ) Q ^ with equivalent norms. By t h i s time i t should be apparent that the above c h a r a c t e r i z a t i o n does not make i t obvious what the. intermediate spaces f o r given "endpoint spaces" are. In many cases the c h a r a c t e r i z a t i o n i s well-known but i n others, i t i s s t i l l an open question and often, what would appear to be the l o g i c a l choice i s wrong (see c o r o l l a r y 3.9.1). However, before e x p l i c i t l y d e f i n i n g the intermediate spaces which we w i l l l a t e r f i n d of i n t e r e s t some standard r e s u l t s i n i n t e r p o l a t i o n theory should be stated. We begin with the following d e f i n i t i o n s . D e f i n i t i o n 1.23 : I f ( A q , A^) and ( B q , B^) are i n t e r p o l a t i o n p a i r s i n A and L r e s p e c t i v e l y , T(A, L) i s the set of a l l l i n e a r maps T from A + A, in t o B + B, such T|A. i s a bounded l i n e a r transformation of o 1 o 1 1 1 A± i n t o . We w i l l denote | | T j | | by M± . - 11 -D e f i n i t i o n 1 . 2 4 : I f A and B are intermediate spaces of A Q , A ^ and B Q , B ^ r e s p e c t i v e l y , we say that A and B have the i n t e r p o l a t i o n property i f f f or every T e T(A, L ) , T | A i s a bounded l i n e a r transformation of A i n t o B . The spaces A and B are then c a l l e d i n t e r p o l a t i o n spaces with respect to ( A Q , A^) and ( B Q , B ^ ) . In p a r t i c u l a r we say that A and B are i n t e r p o l a t i o n spaces of type 9 (where 0 £ 6 < 1) i f | |T|A| I = M < CM^"^ f o r a l l T e T(A, L) where C >^  1 i s independent of T. The i n e q u a l i t y i s said to be exact i f C = 1, Theorem 1 . 2 5 : Suppose ( A Q , A ^ ) and ( B Q , B ^ ) are two i n t e r p o l a t i o n p a i r s of A and L r e s p e c t i v e l y . Then the spaces ( A Q , A^)g and ( B , B , ) . „ for 0 < 9 < 1 , 1 < q < °° and/or 0 < 9 < 1 , q = °° are o 1 9,q;K — — — i n t e r p o l a t i o n spaces of ( A Q , A ^ ) and ( B Q , B ^ ) of type 9 and M <. M^"9M9 (T E T ( A , L)) . Theorem 1 . 2 6 : The intermediate spaces ( A , A , ) . , and ( B , B , ) _ „ v . o 1 9,q;J o' 1 9,q;K fo r 0 < 9 < 1 and 1 <_ q _< °° are i n t e r p o l a t i o n spaces of ( A Q , A ^ ) and ( B Q , BjJ of type 9 . As mentioned before we can extend the intermediate spaces of the P o P l Lebesgue spaces L and L to a la r g e r c l a s s of spaces, namely the Lorentz spaces, which contain the L P spaces as p a r t i c u l a r cases. We begin - 12 -with D e f i n i t i o n 1.27 : Let f be a measurable function. I t s non-increasing (or equimeasurable) rearrangement f i s given f o r x > 0 by f*(x) = i n f { t : D f ( t ) <_ x} , where D^ i s the d i s t r i b u t i o n function of f (see d e f i n i t i o n 1.3). D e f i n i t i o n 1.28 : The Lorentz space L(p, q) i s the space of a l l measurable functions f such that f ' 'p,q 1/q * r [ t i / p f * ( t ) ] q ^ L P i 0 t J < 00 I* J-h**, ~_ q v »» a n a | | L | | 1 <_ p <_ °° and q = for 1 <_ p < °°, 1 <_ q < « and | |f | |* = sup t / p f * ( t ) < «, when P ' q t>0 It should be mentioned that ||•||* i s not always a norm since the t r i a n g l e i n e q u a l i t y may f a i l . We w i l l introduce a norm on L(p, q) (which then provides an a l t e r n a t e d e f i n i t i o n to 1.28), but i t i s much easier to work with " 1 1 "p,q Since | | f * | | p - | | f | | p and ||f||Jjp - | | f * | | p i t follows that L(p, p) = L P . Theorem 1.29 : Let f e L(p, and s e l e c t ^ 2 — q l * T h e n I If II < I]fI| „ and hence L(p, q n) CL L(p, q 9) .. I I ' p,q 2 — p»q 1 1 1 - 13 -D e f i n i t i o n 1.30 : A subadditive operator T i s of r e s t r i c t e d weak type (p, q) i f i t s a t i s f i e s the weak type (p, q) conditions when r e s t r i c t e d to c h a r a c t e r i s t i c functions of sets of f i n i t e measure. Theorem 1.31 : Let T be a subadditive operator of r e s t r i c t e d weak types ( r Q , p ) and ( r ^ , p^) where TQ < r ^ and p Q ^ p^ . Then there e x i s t s B = B(9) such that l l T f l l * <B||f||* 1 1 "p,q - " " r , q ]_ — 0 Q for a l l f e L ( r , q), where 1 < q < 0 0 and — = + — , - - P P Q PjL 1 1 — f) A - = + 2— a n d 0 < 0 < 1 . r r r, o 1 Theorem 1.5 i s a s p e c i a l case of the l a s t theorem but theorem 1.31 i s sharper as i t concerns a larger c l a s s of spaces. A p p l i c a t i o n of t h i s to the Fourier transform y i e l d s a stronger form of the Hausdorff-Young i n e q u a l i t y which w i l l be of l a t e r i n t e r e s t . C o r o l l a r y 1.31.1 : Let f e L P ( R n ) , 1 < p < 2. Then f e L(p', p) (where •+ ^  = 1 ) and there e x i s t s a constant B dependent only on p such that i i<., p i» imi p -D e f i n i t i o n 1.32 : The average function of f i s ** 1 f ( t ) - - f (x) dx 0 - 14 -We now define the norm q IP f 0 0 1/P ** q d t ) 1 / q [ t 1 / p f ( t ) ] q ^ j , where 1 _< p < 0 0 , 1 <_ q < 0 0, and i i l / o ** f I = sup 1: / p f (t) , P ' q t>0 where 1 _< p £ °°, q = °°. Many authors (see [3]) define the L(p, q) spaces i n terms of t h i s norm. However the i n t r o d u c t i o n of f greatly increases the complexity of c a l c u l a t i o n f o r any given function f. In any case the two d e f i n i t i o n s are equivalent as Theorem 1.33 : For f e L(p, q), 1 < p <_ «> , 11*11* < 11*11 < -V llflf 1 1 M p , q - 1 1 1 'p,q - p - i 1 1 M P t q Furthermore Theorem 1.34 : For 1 < p <_ 0 0, 1 < q < °°, L(p, q) with the norm ||•|| i s a Banach space. We also have the following Theorem 1.35 : The spaces L(p, q) (for l < p < ° ° , l £ q < ° ° and/or 1 oo l £ p £ < » , q = °°) are intermediate spaces of L and L . They are - 15 -equal to the spaces (L\ L ) - and | | f | | = 11*11 i i - i , q ; K , p , q i~.q;K A d d i t i o n a l l y f o r 1 < p < 0 0, 1 <_ q _< 0 0, the bounded f u n c t i o n a l on (L 1 , L°°) defined by | | t 1 / p f * ( t ) | | s a t i s f i e s l-±,q;K L Q I t 1 / P f * ( t ) | | < | | f | | < - P T | | t 1 / p f * ( t ) | | . l--,q;K P L q Theorem 1.36 : For 1 ;S P Q < P-^  :£ °°» 1 _£ q £ °°, the spaces L(p, q) are Po P l p o P l the intermediate spaces (L , L ) Q of L and L , where D ,q;K I = i z i + e _ , 0 < e < i . p p o p i Proof : This follows from 1.35 by r e i t e r a t i o n . D e f i n i t i o n 1.37 : Let u be a measurable function. Then we define the weighted L P space L P to be the class of a l l functions f such that the product fu belongs to L P and we define | |f | | ^ = | |fcj| | p . We wish to discuss the intermediate spaces L p , L P % Ve ) P;K L P , L P • uo Ul Je,q;K = L P , with an It I t i s known that the "diagonal spaces" 1—0 0 equivalent norm, where u) = to^ co^  . We mention that Calder6n i n [4] characterized the intermediate spaces obtained by the complex method. They are [ L P , L P ] Q = L P where Q v(x) = u(x) . As before however we are interes t e d i n the " r e a l " - 16 -i n t e r p o l a t i o n methods as they y i e l d more general r e s u l t s . Due to the complexities of the arguments involved we quote from G i l b e r t [6], c o r r e c t i n g a typographical e r r o r , but omitting the proof. Theorem 1.38 : I f 1 <_ p <_ », and 0 < 8 < 1, 1 <_ q <_ °° and r > 1, the expression I k=-°° +ke r k~ 1<w(x)<r k f ( x ) | P dp 1/p] 1/q defines a norm on ( L P , L P)„ ' to 0,q;K More generally L P L P o l-'e.qjK i s the c l a s s of functions f -1. such that the above expression i s f i n i t e where u> = o> a) and du - to dt. o 1 o An equivalent norm on the intermediate spaces i s given by I [[ | f ( x ) - o ( x ) 1 _ e - 1 ( x ) e | P dx k=-~ n - l k r <to(x)<r q/p} 1/q In the case that p = q these reduce to weighted L P spaces. In the general case however, they are c a l l e d mixed norm spaces. The reader i s rederred to Benedek and Panzone [2] for a d e t a i l e d d i s c u s s i o n . CHAPTER TWO LIPSCHITZ CLASSES ON TOTALLY DISCONNECTED GROUPS Let G be a V i l e n k i n group, that i s , a compact, metrizable, O-dimensional, abelian group. Then the dual group X of G i s a countable d i s c r e t e , abelian, t o r s i o n group. V i l e n k i n [11] proved Theorem 2.1 : There e x i s t s an increasing sequence ^ x n ^ °f f i n i t e subgroups of X and a sequence {<l>n} °f characters i n X such that ( i ) X Q = (X q} where X Q ( X ) = 1 f o r any x e G ; ( i i ) f o r n >_ 1, x n / x n _ i i s o f prime order p n ; ( i i i ) X = (J X : n=0 n (iv) <j> e X . _\ X f o r a l l n > 0 ; n n+1 n — Pn+1 (v) d> e X f o r every n > 0 . v T n n — Next we w i l l enumerate the elements of X by means of the $ . n s Set m = 1 and define m. = J J " p . . Now, i f k >_ 1 and i f k = [ i i i = l i=0 with 0 < a. < p . , , f o r 0 < i < s define — l l + l — — a a , o s xk " ^o ^s * Then we can write X = { Y . : 0 < i < m } . Next l e t n A i — n G n = { x e G : x f e(x) = 1, a l l x k e X R } - 18 -be the a n n i h i l a t o r of X . I t i s cl e a r that n G = G " D G. 3 . . . O n G = {0} o 1 ' ' n n=0 and that the G s form a fundamental system of neighbourhoods of zero i n G. Furthermore, f o r each n > 0 there e x i s t s x e G \ G ,, such that ' — n n n+1 X (x n) = exp (2-rri/p n +^) and each x e G can be uniquely represented i n n the form x = Y b.x. where 0 < b.< p.., f o r every l > 0 . . n i i — i i+1 J — i=0 A d d i t i o n a l l y 00 G = { x e G : x = Y b.x., b = . . . = b = 0 } . n . L n l l o n-1 i=0 Consequently each coset of G^ i n G can be represented by z + G n where n-1 z = Y b.x. f o r some choice of 0 < b. < p.., . We s h a l l order these z 1-0 1 1 " 1 1 + 1 l e x i c o g r a p h i c a l l y and denote them by z where 0 < q < m q>n - n If dx i s the normalized Haar measure on G, the Fourier s e r i e s of f e L^(G) i s the s e r i e s S[f](x) = I f ( n ) x (x) n=0 where f(n) = The p a r t i a l sums are given by f ( t ) x _ ( t ) dt G n - 19 -k-1 A S k ( f , x) = I f(n ) x n(x) = n=0 f(x-t)D, (t) dt G K k-1 where ^ ( t ) = I x n ( t ) ^ s t n e D i r i c h l e t kernel of order k. The n=0 D i r i c h l e t kernels have the property that D m (x) - { n 0 i f x i G , n m i f x e G ; n n ' (see V i l e n k i n [11]), For the remainder of t h i s discussion G and X w i l l be as described above with the f u r t h e r r e s t r i c t i o n that sup p = p < 0 0 . n (The usual example of a group of t h i s type i s given by G = TT z(pn> , n=l where ^P n^ i - s a bounded sequence of prime numbers (not n e c e s s a r i l y d i s t i n c t ) . Then x = (b , b,, .... b , ...) where b. = 0 i f i ^ n and n o 1 ' n I b n = 1, and i f x e G i s a r b i t r a r y , f n ( x ) = e x P ^^^n^n+l^' *n t n e case that p n = 2 f o r every n, G i s merely 2 W and the elements of the character group X are the Walsh functions.) We need the following d e f i n i t i o n s : D e f i n i t i o n 2.2 : For B > 0 we define A(B) to be the set of a l l i n t e g r a -b l e functions f on G such that - 20 -I |f (n) | 3 < ~ for g < °° , n=0 sup I f (n) | < °° for 8 = °° ; n 8 i . e . we say f e A (B ) i f f f e I D e f i n i t i o n 2.3 : Let 1 <_ p <_ «>, k E N. Then for f e L P(G) we define the integrated modulus of continuity of order k to be u) p(f, k) = sup{||T yf - f | | p : y e G k } , where T y f ( x ) = f ( x + y) . D e f i n i t i o n 2.4 : I f 0 < ct <_ 1, 1 <_ p <_ <*>, l<_q<_°°, then we define Li p (a, q; L P ) to be the set of a l l f e L P such that {m^u p(f, k) }fc c &q and we define a norm on t h i s space by f 0 0 ^ l / q l | f | | T . r T P ^ = M f l l + I (f» k>>q 1 1 M L i p ( a , q; L ) 1 1 " p [ k ^ Q ^  p v with the appropriate modification i f q = 0 0 . Some authors (for example V i l e n k i n [11] and Onneweer [7, 8]) 00 consider a space L i p a which i n our notation i s Lip(ct, °°; L ). The classes CO L i p ( a , q; L ) are used to deal with Fourier m u l t i p l i e r s i n [5] and [12]. I t i s c l e a r that f o r 1 <_ p £ 0 0, 1 <_ q, s _< 0 0, and 0 < ct-^  < £ 1> w e have LipCc^, q; 1 P) CL.Lip(a^, s; L P ) . Furthermore, i f 0 < ct < 1, l < q < o o and 1 < p < p, < « then — — — — o — 1 — - 21 -P l P o L i p ( a , q; L ) CL L i p ( a , q; L ) and l a s t l y , i f 0 < a <_ 1, 1 <_ p £ °°, and 1 _< q Q _< q^ < oo then Lip (a, q Q; L P ) CL L i p (a, q^; L P ) D e f i n i t i o n 2 .5 : I f f i s a function on G and H CL G, then we set os c ( f , H) = sup { |f(x) - f ( y ) | : x, y e H } . The following theorem i s due to Onneweer [7] and i s of i n t e r e s t to us mainly because of i n e q u a l i t i e s (1) and (2) i n the proof. Theorem 2.6 : I f f e L r ( G ) , 1 < r < 2 and i f °° r ™k ^ 11/r I I (osc(f, z k + G k ) ) r < » , k=0 *• q=0 q,k K J then f e A ( l ) . Proof : For every k ±_ 0 the Fourier s e r i e s f ( x + x^) - f(x) i s 00 I f(n)(x n(x k) - Dxn(x) . n=0 k For _< n < n^^) w e have n = £ a±m± w i t n u 1. a± < V±+i f o r a 1 1 * i=0 and / 0 . Now as x^ e G^, i t follows that X n ( x k ) = exp(2Tria^/p^ +^) implying lxn(xk) — 11 ^ _ ^ P ^ ^ _^ TTP ^ . Hence the Hausdorff-Young i n e q u a l i t y implies that i f k > 0 and s i s the index conjugate to r , then 22 -(1) P I | f ( n ) | S  n = m k . "k+i - l n = m k I | f ( n ) | S | x n ( x k ) - l | S 1/s < I I |f(x + x, ) - f ( x ) | r dx G k 1/r r I V 1 t I | f ( L q=0 JG. x + x. - z , ) - f ( x k q,k' r V 1 1 ^ 1 [ [ Q \ <° S C< f' " ^ k + V ^ J 1/r An a p p l i c a t i o n of Holder's i n e q u a l i t y y i e l d s (2) I | f ( n ) | n = m k -1 - ( m k + l " m k ) 1 / r I | f ( n ) | S 1 n = m k 1/s , ,1/r p -1/r i ( m k + l " m k ) 7 "k -1 X (osc(f q=0 Hence I | f ( n ) | < C I n=l k=0 -1 £ (osc(f, z q=0 which i s f i n i t e , i . e . f e A ( l ) - 23 -The following theorem i s also due to Onneweer [8], Theorem 2.7 : Let 1 < p < 2 and 0 < 8 <_ p 1 . I f f e L P(G) and i f k=0 k (p'-B)/p' ( u) p(f, k)) < oo , then f e A(B) . Restated i n our notation t h i s becomes Theorem 2.7' : Let 1 < p < 2 and l < B < p ' . If f e L i p ( i — , , B; L P ) , ; — — — — P P then f e A(B) Proof : F i r s t l e t 1 < p' <_ 2. Then by i n e q u a l i t y (1), for a l l k >_ 0 (3) r "W 1 . n. \W I | f ( n ) | P  n = m k < C I J f (x + x, ) - f (x) | P dx 1/p < Ceo (f, k) Holder's i n e q u a l i t y then y i e l d s " k + r 1 ^ I l*<*>l 8 -1 I. | f ( n ) | P  n = I \ , , 8 / P V "k+i 1 ^CP'-JO/P* I 1 v n ^ > 24 -< C ( _ p ( f , k)) (m^+1 - m k ) ( p , " e ) / p ' Hence I | f ( n ) | P < C ^ - ^ ^ ' - ^ ^ ' ( ^ ( f , k ) ) S n=l x k=0 < C W P " p " p (a) ( f , k ) ) P < - ; k=0 * P i . e . f e A(3) . If p = 1 then p' = 0 0 and the above proof holds with the appropriate modifications. S e t t i n g 3 = 1 i n theorem 2.7' y i e l d s C o r o l l a r y 2.7.1 : Let 1 < p < 2. I f f e L i p ( j , 1; L P ) then f e A ( l ) Co r o l l a r y 2.7.2 : Let 0 < ct <_ 1, 1 < p < 2, and $ > p/(ctp + p - 1). I f f e L i p ( a , °°; L P ) then f e A(g). p 1 — 8 Proof : Note that our assumptions imply a > ,— . Hence Li p (a, °°; L P ) C L i p ( P - T Z T , g; L P ). The r e s u l t follows by 2.7' . pp Choose 3 = 1 i n c o r o l l a r y 2.7.2 and obtain C o r o l l a r y 2.7.3 : Let 1 < p < 2 and p 1 < a £ 1. If f e L i p (a, 0 0; L P ) then f e A ( l ) . - 25 -In [8] Onneweer shows that the l a s t two c o r o l l a r i e s are best possible as there does e x i s t an f e L i p ( a , °°; L P ) which does not belong to A(p/(ap+p-l)) , (1 < p <_ 2, 0 < a <_ 1). This w i l l be discussed i n d e t a i l i n chapter 3. Theorem 2.8 : I f 1 £ p £ 2, 0 < a _< 1 and f e Lip(ct, 1; L F ) then f o r CO a l l 8 > I - a , I n " e | f ( n ) | n=l Proof : An a p p l i c a t i o n of Holder's i n e q u a l i t y y i e l d s I | f ( n ) | 1 ( m ^ ) n = m k -1 1/P I |f(n)T n = m k 1/P' c < W 1 / p V f » k> by (3). Then we have I * nf(n)| < C 1(m l c 4. 1) P « ( f , k) n = m k i-3 < 03 ( f , k ) Now, i n the fundamental case where 8 = 1/p - a, we have I n _ i i | f ( n ) | < C 2 I n£u> ( f , k) < n=l as f e L i p ( a , 1; L ) I f 8 > — - a then P co _ i I n " B | f ( n ) | < I n p | f ( n ) | < •». n=l n=l As a c o r o l l a r y to t h i s we obtain a r e s u l t due to Onneweer [7], C o r o l l a r y 2.8.1 : I f f e L i p (a, °°; L ) f o r 0 < a _< 1 then, for a l l OO $ > i - a , I n ~ 3 | f ( n ) | < » . " n=l Proof : Let B > y - a and set a ± ~ \ ~ ^ . Then oo 2 2 L i p ( a , °°; L ) C L i p ( a , .»; L ) C I L i p ( a ^ , 1; L ) Because < a . The r e s u l t follows.by theorem 2.8. Notice also that s e t t i n g a = 1/p i n theorem 2.8 gives an a l t e r n a t e proof of c o r o l l a r y 2.7.1. We next look at some r e s u l t s of a s l i g h t l y d i f f e r e n t nature. Theorem 2.9 : Let 1 <_ p <_ 2 and 0 < a <_ 1. I f {C n} i s a sequence for which £ |C n a|P < <=° , then there e x i s t s an f e L i p (a, °°; I? ) n=l n such that f(n) = C for a l l n > 1. n — Proof : As £ |n aC | P < «> then £ |C | P < «> . By the Hausdorff-Young n=l n=l theorem there e x i s t s an f e L such that f(n) = C for each n > 1. Next n — - 27 -choose any nat u r a l number q and any y e G . Now Xi_(y) = 1 f ° r q K 0 < k < m and hence - - q T f(x) - f(x) ~ I f ( k ) ( X . (y) - D x v(x) k=m Applying the Hausdorff-Young i n e q u a l i t y y i e l d s T f ' y f M p . l I | f ( k ) ( x k ( y ) - D | p k=m q t °° ^ l / p 2 P I | f ( k ) | P ' k=m q I / P < 2 ^k= ap m m apj 1/P < 2m I | f ( k ) k a | P k=m q 1/p —a < Cm - q Thus f e L i p ( a , °°; ~L ) . Theorem 2.10 : Let 1 <_ p £ 2 and 0 < B < a <_ 1 . Suppose that f e Lip(B> p'; L P ) . Then I | f ( n ) n 6 | P ' < » . \ + l _ 1 Proof : By i n e q u a l i t y (3) we have £ | n e f ( n ) | P <_ C(m^o3 ( f , k ) ) P n = m k P Summing over k y i e l d s the desired r e s u l t . - 28 -C o r o l l a r y 2.10.1 : Let 1 < p _< 2, 0 < B < ct <_ 1. I f f c L l p ( a , °°; L 1 ) then J| | f ( n ) n e | P < «? . n=l Proof : I f B < ct then L i p ( a , °°; L P ) i s a subspace of Lip(B, p 1 ; L P ) . The r e s u l t follows by Theorem 2.10 . We mention at t h i s point that theorem 2.10 i s best p o s s i b l e since there e x i s t s an f e Lip(B, q; L P ) for any q > p' such that oo t £ | f ( n ) n ^ | P = oo . C o r o l l a r y 2.10 i s also best possible since there i s an f e L i p (a, oo; L P) such that £ | f ( n ) n a | P = oo . The reader i s n=l refered to chapter three f o r d e t a i l s . - 29 -CHAPTER THREE APPLICATIONS OF INTERPOLATION THEORY This chapter begins with a b r i e f d iscussion of the intermediate spaces ( L i p ( a , r ; L P ) , Lip(8 , s; L P ) ) . A f t e r determining what these 9 ,q spaces are we examine the theorems of chapter 2 i n the context of interpo-l a t i o n theory. In t h i s manner we extend the r e s u l t s we already have. We also show that these new r e s u l t s are sharp and we give a counterexample i n the i n t e r p o l a t i o n theory of the spaces f Po P l 1 Lip ( c t , r ; L ), L i p ( B , s; L ) . The following i s a g e n e r a l i z a t i o n of work of Fournier [5]. r r We begin by considering (L , L i p ( l , °°; L )) where 1 < r < » t) ,q — — and 0 < 0 < 1, 1 £ q £ ~ . We define r a>r(f, 0) i f t> 1, " ( f ) ( t ) = • I u>r(f, k) i f m^1 <_ t < ^ I t follows that f e L i p ( a , q; L r ) i f f f e L r and t a u ( f ) ( t ) £ L q and the L* norm of t a o ) ( f ) ( t ) i s equivalent to the L i p ( a , q; L r ) norm of f. On the c i r c l e group T we have " ( f ) (s + t) £ c o ( f ) ( s ) + c o ( f ) ( t ) . However t h i s does not hold on our t o t a l l y disconnected group G (see [1]). Hence we introduce - 30 -w(f)(s) 0 J * ( f ) ( t ) = sup s>t S u* has the property that u * ( f ) ( s + t) £ W * ( f ) ( S ) + w*(f)(t) . Replacing w by u* gives the same spaces L i p ( a , q; L ) with equivalent norms. We proceed much as i n [3]. In the r e a l i n t e r p o l a t i o n method, l e t r r A q = L , = L i p ( l , °°; L ) . Then we obtain Lemma 3.1 : I f f e L , 1 < r < », we have ( i ) K(t, f) < (1 + p ) o j*(f)(t) + min(l, t ) | | f | | r , ( i i ) _ * ( f ) ( t ) £ 2K(t, f) , ( i i i ) m i n ( l , t ) | | f | | r < K(t, f) . Proof : Write f = f + f, with f e A , f. e 1 . Then o 1 o o' 1 1 u)*(f)(t) <_ o j*(f Q) + uMfp < 2||f || + t s u p ^ H l I o r r s s>t < 2 | | f o | | r + t | | f 1 | | L i p ( l , »; L r ) since oi(f) (s) f J. , ,. sup = sup m, (o ( f , k) . s . ic r s k In a d d i t i o n - 31 -mm (1. t ) | | f | | r < m i n ( l , t ) ( l | f o | | r + Mf-Jlj.) l l|f 0 l lr + t l l f l l l L i p ( l , <-; L r ) Passing to the infimum over a l l p a i r s f Q , with f Q + = f , we obtain i n e q u a l i t i e s ( i i ) and ( i i i ) . To prove ( i ) , i f t >_ 1 write f = fQ + 0 . Then K(t, f) < | | f | l + t||0|I L i p ( l , »; L r ) = min(l, t)|If||r . Now f o r t < 1, choose k such that m^ <_ t £ m^i_ ™k f-^x) = I f(n)x n(x) n=l Let and l e t f (x) = f(x) - f. (x). Then f, = f*D . As MD I L = 1 and o J. x m^ 1 1 m^ ' 1 1 D =0 o f f G, , i t follows that I If II < OJ ( f , k) , but m^ k 1 1 o l l r — Vs u r ( f , k) = o j(f)(t) < _ o)*(f)(t) . Also | |f | | r <_ | | f | | . Furthermore, o)(f 1)(s) i s bounded above by o)(f) (s) and ||f||^w(D ) ( s ) . I t follows m k from the second bound that w ( f ± ) (s) vanishes on 0 < s < m^ because o)(D ) (s) also vanishes there. Hence m k " ( f l ) ( s ) e Q)(f)(s) co(f)(s) sup < sup •'• <_ p sup . _^ S i s s s -1 s>t s ^ k Therefore - 32 -K(t, f) 1 | | f 0 | | r + t | \ \f1\\T + sup i | < u * ( f ) ( t ) = t | | f | | r + p u * ( f ) ( t ) < (1 + P ) u * ( f ) ( t ) + min(l, t ) | | f | | r which completes the proof of the i n e q u a l i t i e s . —8 Now ( i ) and ( i i ) imply that the map t — > t K(t, f) belongs to L* i f f t — > t - 9 n ) * ( f ) (t) belongs to L q . This, together with our e a r l i e r q * comment where we characterized membership i n L i p ( a , q; L ) i n terms of membership of t a u * ( f ) ( t ) i n L q proves Lemma 3.2 : Let 1 < p _< », 1 <_ q £ °°, 0 < 9 < 1. Then ( L P , L i p ( l , oo; L P ) ) = Lip(0, q; L P ) . 9,q Applying the theorem of r e i t e r a t i o n (theorem 1.22) to lemma 3.2 y i e l d s Theorem 3.3 : Let 1 <_ p <_ °°, 1 <_ r , s <_ and 0 < a, 3 < 1. I f 0 < 8 < 1, l £ q _ < o o and a £ $ then the space (Lip(a, r ; L P ) , Lip($, s; L P ) ) 0,q i s L i p ( y , q; L P ) where y = (1 " 9)a + 88 - 33 -D e f i n i t i o n 3.4 : A(B, q) i s defined to be the cl a s s of a l l integrable functions f such that f belongs to £(8, q) (where £(8, q) i s the d i s c r e t e analogue of L(8, q) ). The reader should note that i f 1 £ 8-^  < 8 2 £ °°, 1 £ r £ °°> then A(6 1, r) O A ( 8 2 , r) ; also i f 1 £ 8 £ °°, 1 £ r £ s < », then A(8, r) C A(B, s) . Thus we r e a l i z e that A ( l ) = A ( l , 1) d A(8, q) f o r any 1 £ 8 £ °°, 1 £ q £ 0 0 . In a d d i t i o n , i f a ^ 8 , we have (A(a), A(8)) f l „ = A ( ( l - 9 ) a + 98, q) , 0,q with equivalent norms. We next consider Theorem 3.5 : For 1 £ P £ 2, 1 £ q £ <=° and 0 < 9 < 1, define 8 by 1 1 - 0 . 8 8 = ~V + 1 * T h 6 n L i p ( | , q; L P ) A(8, q) . Proof : By 1.31.1 we have L P d A(p', p) and by s e t t i n g a = 1/p i n 2.8 we have Lip(pS 1; L p ) d A ( l ) . In t e r p o l a t i o n y i e l d s - 34 -( L P , L i p ( | , 1; L P ) ) Q j q Cl (A(p», p), A ( l ) ) 6 ) q , 8 p 1 1~0 9 i . e . L i p ( - , q; L p ) C A (8 , q) where - = — 7 - + "r . P ts p i Set q = 8 i n theorem 3.5 and obtain C o r o l l a r y 3.5.1 : I f 1 < p < 2, 0 < 6 < 1 , and 8 i s given by 1 = i = i + f » then Lip(£, 8; L P ) CL A(0) . P P -L P I f we set 6 = £ - ^ and r e s t r i c t 8 to the i n t e r v a l 0 < 8 < p'. 8 p we r e a l i z e that t h i s i s exactly theorem 2.7' with the exception of the endpoint 8 = P 1 • Hence we see that given Hausdorff-Young and the endpoint r e s u l t 2.8 (or 2.7.1), we get theorem 2.7' by i n t e r p o l a t i o n . As a bonus, we get a d d i t i o n a l r e s u l t s concerning i n c l u s i o n i n A (8 , q) . Theorem 3.6 : For 0 < a < 1, l £ P £ 2 , l £ q £ ° ° and 8 > p/(ctp + p - 1) we have Lip (ct , q; L P ) C L A ( 8 ) . Proof : As a > 77 - ^7- the r e s u l t follows by 2.7' . 8 P Theorem 3.6 can also be proven using 2.7.2 and a simple i n t e r p o l a t i o n argument. Theorem 3.7 : I f l < p < 2 , l < q < ° ° and p * < 8 < 1, then any f e Lip ( 8 , q; L P ) also belongs to A(l) - 35 -Proof : As 8 > — we have Lip(8, q; L P ) C L i p ( ; k 1; L P ) CZ. A ( l ) by p P 2.7.1 . We wish to show that the l a s t two r e s u l t s are, i n some sense, best poss i b l e . (Onneweer did t h i s i n the s p e c i a l cases given by c o r o l l a r i e s 2.7.2 and 2.7.3). We begin by d e f i n i n g R, (x) = D (x) - D (x) f o r fc "he V - l k > 1. Now i f x e G, _ m k 0 i f x e G\G. i f x e G ^ N G k Jk-1 and f o r p > 1 we have G. AG, k-1 k G\G l \ ( x ) Jdx k-1 -1 - m ^ - m^J* + ( m ^ - n ^ 1 ) ^ . ! p-1 . p-1 1 \ + V i <_ CmP 1 . Next f o r 1 < p £ 2, 0 < a <_ 1, S >_ 0 define -(a+l-p ) °o I I L h n ,(x) = I -- R. (x) - 36 -C l e a r l y h . e 1/ as 1 a,p , 6 ''p - ^ r -(a+l-p X ) H\Cx)||P< Now for every n, i f y e and k <_ n then R^Xx + y) = R^ x ) . Hence |T h " h II 1 y a,p , 6 a , p , 6 ' ' p -(a+l-p" 1) • I NT^-^M k=n+l k 6 y < I -(a+l-p x ) \ k=n+l - ( k + i - p - 1 ) _ 1 <c I —* \ P k=n+l k° * = C I -a k=n+l k _ „ -a - 6 < Km n , — n since {m^} grows geometrically. Hence X f k=l ^ P ~ k=l " " I k k=l -<Sr - 37 -which i s f i n i t e f o r 6r > 1 . I f we put 6 = 0, then ^ ( f , n) <_ mV* . Consequently {m w ( f , n)} e I and h . £ L i p ( a , °°; L ) . 7 n p ' n a,p,6 If we s e l e c t 6 > 0, then, i f r > 1/6 , we have h x e L i p ( a , r ; L P ) a,p,6 Fur thermore, i t i s c l e a r that f o r £. n £. ^  » -(a+l-p" 1) (4) h. „ ,(n) = ^ a,p,6 ,6 Thus n - l I p/(ap+p-1) n n ^ ( a + 1 _ P 1)-p/( aP+P -D ( m k " m k - l ) k6-p/(ap+p-1) -1 n J x ( m k " m k - l ) k6-p/ (ap+p-1) k=l ™k k^P/(aP+P - D which c l e a r l y converges i f f 6 > (ap+p-1)/p . In general h „ e A(B) for any 8 > p/(ap + p - 1) and a,p, o h . e A(p/(ap + p -1)) i f f 6 > (ap+p-1)/p . a,p,6 Thus we have - 38 -Theorem 3.8 : I f 1 < p < 2, 0 < a £ 1 and r > p/(cxp + p - 1) and 5 i s chosen such that p/(ap+p-1) < — < r the function *\x p <5 e ^P^ 0 1' r ' L ^ k u t i s not i n A(p/(ap+p-1)). In p a r t i c u l a r , f o r r > 1, the function h^ e L i p ( — , r; I?) but i t s Fourier s e r i e s -,P,1 P i s not absolutely convergent . Theorem 3.5 can also be seen to be sharp; that i s , i f s < q, then f o r s u i t a b l e choice of a, p, 6, the function h „ belongs to * a,p,6 L i p ( a , q; I?) but does not belong to A(g, s ) . We state the following theorem : Theorem 3.9 : Let l < r < 2 , 0 < 9 < 1, 1 £ q _< °° . Then ( L i p ( | , 1; L 2 ) , L i p ( p 1; L r ) ) 0 j q C.A(1) . Proof : In c o r o l l a r y 2.7.1 we make the following s u b s t i t u t i o n s : 1 2 p = 2, hence LipOj, 1; L ) C A ( 1 ) , 1 r p = r , hence L i p ( — , 1; L ) C Z A ( l ) . I n t e r p o l a t i n g between these statements gives the desired r e s u l t . The obvious question to ask at t h i s point i s what exactly are the 1 2 1 r spaces (Lip(-«, 1; L ), L i p ( — , 1; L )) . In l i g h t of what we already z r B,q known about the intermediate spaces (L °, L ^) and the spaces 9 ,q - 39 -(Lip(ct, r ; L p ) , Lip(B, s; L P ) ) i t would seen reasonable to expect that 9 ,q f o r 0 < 9 < 1, 1 < q < » (Lip(^-, 1; L 2 ) , L i p ( - , 1; L r ) ) Q would be — — z r 0 ,q 1 1 1—9 9 the space L i p ( — , q; L(p, q)) where — = — 1 . However, t h i s i s not P P ^ ^ the case. In f a c t , i f we consider the case q = p we see C o r o l l a r y 3.9.1 : I f 0 < 9 < 1, (Lip(-k 1; L ), L i p ( - , 1; L r ) ) Q does r. 1 r °>P not contain L i p ( — , p; L P ) where — = •^p- + — . In f a c t , i f 1 < q <_ <» , P • P ^ i-LipO^, q; I?) i s not contained i n the intermediate space i n question. Proof : Consider the function h r already defined. For f i x e d q > 1, a,p,6 s e t t i n g 6 = 1 produces a function i n LipC^j-, q; L P ) which i s not i n A ( l ) and hence we see L i p ( a , q; L P ) i s not even contained i n the intermediate space we d e s i r e . As was mentioned i n chapter 2, theorem 2.10 and c o r o l l a r y 2.10.1 are best p o s s i b l e . In f a c t Theorem 3.10 : Let l < p < _ 2 , 0 < a <_ 1, r > p ' . Then there e x i s t s an f E L i p ( a , r ; L P ) such that I |f(n)n n=l a i P 1 Proof : Consider the function h „ already defined. I f 6 = 0 , i t a,p,6 belongs to L i p ( a , °=; L P ) . For any other r > p' s e l e c t 6 > 0 such that p' < 7 < r . We then have h £ L i p ( a , r ; L P ) . Furthermore, by (4), 6 ct»P»o - 40 -we see co V 1 I I |h a 6 ( n ) n k=l n=mk_ ' P ' a,p' "Is " k-1 ( m k " n k - - ) -(a+l-p ) a p» r x \ "k ' 1 c I k=l l-ap'-p'+p'/p+ap1 as 5p' < 1 D e f i n i t i o n 3»11 : Let 1 £ p £ °°, l £ q < _ ° o . Then the mixed norm space &P i s defined to be the space of a l l sequences (C } such that a n n ,q I k=0 r I a_ ip ) n C F u 1 n 1 2 k<n<2 k + 1 q/p 1/q < oo i f 1 <_ p, q < 0 0 ; with the appropriate modifications i f p or q i s i n f i n i t e . One can show that for integrable functions f on G, f £ £ a n ,q i f f I k=0 , "k+r 1 I | n a f ( n ) | P  L n = m k q/p 1/q < 00 - 41 -and t h i s produces an equivalent norm. Hence Theorem 3.12 : Let 1 <_ p <_ °°, 0 <_ a Q , < » . Then i f 0 < 6 < 1, 1 < q < oo t the space £ P , *P } a a, I n ° vv 1) = I 9,q n ,q where B = (1 - 6)a Q + 90^  C o r o l l a r y 3.12 : Let l £ p £ ° ° , 0<e<l, l £ q £ ~ . Then £ P, Z 2 n ,°^9,q n ,q Proof n Now r e i t e r a t e . l/2,< Theorem 3.13 : For 1 < p < 2 and 0 < 9 < 1, i f f c Lip (9 , q; L P ) then f e % 9 n ,q i Proof : By Hausdorff-Young, ( L P ) * CL lV and by i n e q u a l i t y (3) we n have L i p ( l , 0 0; L P ) A CL £ P ^ . I n t e r p o l a t i o n y i e l d s the desired r e s u l t . n , 0 ° As an immediate c o r o l l a r y , we obtain theorem 2.10. We have already seen theorem 2.10 i s sharp. An analogous argument to that used i n theorem 3.10 w i l l show that 3.13 i s sharp; that i s , i f r > q and we s e l e c t 6 i n the i n t e r v a l (—, — ) , the function r q - 42 -h . E L i p ( 6 , r ; L F ) but h Q . does not belong to & H , 9,p,6 0,p,6 F i n a l l y Theorem 3.14 : I f 1 < p < 2, 0 < 9 < 1, 1 < q <_», then i f { (M i s a sequence i n &P , there e x i s t s an f belonging to L i p ( 6 , q; L P ) o n ,q such that f(n) = C for each n > 1 . v n — Proof : Interpolate between the statements *p cz (L PV o n and a p 1 CL L i P ( l , co; 1 / ) - . n ,°° - 43 -BIBLIOGRAPHY 1. L.A. Balashov and A.I. Rubinshtein, "Series with respect to the Walsh System and Their Generalizations", Journal of Soviet Mathematics 1 (1973), 727-763. 2. A. Benedek and R. Panzone, "The space L P with mixed norm", Duke Math. J . 28 (1961), 301-324. 3. P. Butzer and H. Berens, "Semigroups of Operators and Approximation", Springer-Verlag, New York, 1965. 4. A.P. Calder6n, "Intermediate spaces and i n t e r p o l a t i o n , the complex method", Studia Mathematica 24 (1964), 113-190. 5. J.F. Fournier, " M u l t i p l i e r s of weak type", Unpublished Manuscript 1974. 6. J.E. G i l b e r t , " I n t e r p o l a t i o n between weighted L P-spaces", Arkiv f5r matematic, 10 (1972) NO. 2. 7. C.W. Onneweer, "Absolute convergence of Fourier s e r i e s on c e r t a i n groups", Duke Math. J . 39 (1972), 599-609. 8. C.W. Onneweer, "Absolute convergence of Fourier s e r i e s on c e r t a i n groups, I I " , To appear, Duke Math. J . 9. E.M. Stein and G. Weiss, "Introduction to Fourier a n a l y s i s on Euclidean Spaces", Princeton U n i v e r s i t y Press, Princeton, 1971. 10. E.M. Stein and G. Weiss, "An extension of a theorem of Marcinkiewicz and some of i t s a p p l i c a t i o n s " , J . Math. Mech. 8 (1959), 263-284. - 44 -11. N.Ja. V i l e n k i n , "On a c l a s s of complete orthonormal systems", Amer. Math. Soc. T r a n s l . 28 (1963), 1-35. 12. M. Zafran, " M u l t i p l i e r Transformations of Weak Type", To appear, Annals of Math. 13. A. Zygmund, "Trigonometric S e r i e s " , 2nd E d i t i o n , V o l . I and I I , Cambridge U n i v e r s i t y P r e s s , 1959. 

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