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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 S t a t e 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 t h e Department of Mathematics  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1973  In p r e s e n t i n g  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 r e q u i r e m e n t s f o r  an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  study.  copying of t h i s  be g r a n t e d by the Head of my  Department or  I t i s u n d e r s t o o d t h a t c o p y i n g 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 g a i n s h a l l not be a l l o w e d w i t h o u t written permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  \JU  g  thesis  my  ii  ABSTRACT  The s u b j e c t o f t h i s t h e s i s i s t h e f r a c t i o n a l o r d e r S o b o l e v s p a c e , Hp, as c o n s i d e r e d b y N i k o l ' s k i i ;  t h e g o a l i s t o demonstrate an imbedding  theorem f o r Hp* analogous t o t h e c l a s s i c a l imbedding theorem f o r W*^ w h i c h was f i r s t shown by S o b o l e v . r The p r o p e r t i e s e s t a b l i s h e d h e r e f o r spaces Hp d e f i n e d o v e r a l l o f R , i n c l u d i n g completeness and imbedding theorems, a r e demonstrated b y n  a t e c h n i q u e i n v o l v i n g t h e a p p r o x i m a t i o n o f f u n c t i o n s i n t h o s e spaces b y entire functions o f the exponential 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 ,  w h i c h a r e o f i n t e r e s t i n t h e i r e own r i g h t , a r e d e v e l o p e d i n a s e p a r a t e chapter.  An e x t e n s i o n 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 d e f i n e d o v e r  an ©pen s u b s e t o f R  n  i s also proved.  iii  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 . .  CHAPTER FIVE  APPROXIMATION OF CONTINUOUS FUNCTIONS BY  CHAPTER S I X  19  ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE . .  30  THE NORMED SPACE  U3 P -••  CHAPTER SEVEN  AN IMBEDDING THEOREM'FOR THE SPACE H£ . . .  53  CHAPTER EIGHT  THE CLASSES HJJ(G ) , w £ ( O n ) , B^( G )  62  BIBLIOGRAPHY  66  n  Q  ACKNOWLEDGEMENTS  I wish t o express my g r a t e f u l appreciation to Dr. R. A. Adams, my thesis supervisor, f o r introducing me to Soholev Spaces, and f o r his continual assistance, guidance, and encouragement writing of t h i s t h e s i s .  during the  I also wish t o thank Dr. J . Foumier f o r h i s  help and assistance i n reading t h i s work. The f i n a n a c i a l assistance of the University o f B r i t i s h Columbia and the National Research Council i s also acknowledged.  1  CHAPTER ONE INTRODUCTION  The S o b o l e v space ^ ( of R  11  G n  ) o f 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  and h a v i n g t h e r e 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 o r d e r m i n -  t e g r a b l e t o t h e power p >_ 1, i s w e l l known, and a g r e a t many r e s u l t s c o n c e r n i n g S o b o l e v spaces a r e b e i n g c o n t i n u a l l y f o r m u l a t e d .  We n o t e , however,  t h e i n c o m p l e t e n e s s o f t h i s c o n c e p t b y t h e r e s t r i c t i o n t h a t t h e parameter m be an i n t e g e r .  I n t h i s work we examine a S o b o l e v space o f f r a c t i o n a l  o r d e r and i t s p r o p e r t i e s . We g i v e a s u r v e y o f some o f t h e many papers b y S. N. N i k o l ' s k i i on t h e space ^ ( G ^ ) where r i s a l l o w e d any r e a l v a l u e . i n v e s t i g a t e a l l p o s s i b l e imbeddings  I n p a r t i c u l a r we  f o r t h e space H^, and examine t h e r e -  l a t i o n s h i p between t h i s and t h e t r a d i t i o n a l S o b o l e v space i n t h e case r = m an i n t e g e r . The t e c h n i q u e 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 t h e a p p r o x i m a t i o n o f c o n t i n u o u s 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 t h e e x p o n e n t i a l t y p e .  We  f i r s t g i v e t h e c o n d i t i o n s f o r t h e e x t e n s i o n o f a f u n c t i o n f e H ( G ) from p  G  n  t o t h e e n t i r e space R , and s u b s e q u e n t l y t r e a t o n l y t h e c l a s s H ( B ) . n  r  n  We n e x t d i s p l a y some r e l e v a n t p r o p e r t i e s o f e n t i r e f u n c t i o n s o f e x p o n e n t i a l t y p e , and t h e n g i v e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e approxi m a t i o n o f a f u n c t i o n f b e l o n g i n g t o HptR ) b y such e n t i r e f u n c t i o n s ; t h e 13  d e m o n s t r a t i o n o f a l l subsequent r e s u l t s i s b a s e d on t h i s a p p r o x i m a t i o n . Completeness  i s g i v e n f o r H ^ R ) w i t h a u s u a l norm, and t h e p r i n c i p a l 11  r e s u l t o f t h i s work, t h e imbedding theorem, i s p r o v e d i n Chapter Seven.  2  F i n a l l y , b r i e f mention i s made of the spaces w £ and Bp where the parameter r i s again allowed any p o s i t i v e r e a l value.  We note that the results given here are not p r e c i s e l y of the same form as dealt with by N i k o l ' s k i i .  In p a r t i c u l a r the d e f i n i t i o n  of the space Hp* presented here i s not the usual one.  We have made  some changes both i n d e f i n i t i o n s end i n proofs i n order to f u l l y elaborate 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 l i n e a r space and f e M, then the norm of f i n M i s denoted by | | f : Gn  M||.  denotes a open domain i n R n  i s the set o f points i n Gn  n dimensional Euclidean space; Gn  ^  at distance greater than n from the boundary of  G ; G* _ i s the set of points i n G at distance less than l / n from the n' n,n * n,n origin. (k) 3x  I f f = f ( x , * * , x ) i s a function defined on G , ,  1  n  n  then f£  i  '  and Aj.(f,h) =  f(x  l t  «*' 3tn). t  A ^ ( f , h ) = ^ ( f . h ) - fU-L/'-.Xi + h,***,x ) - fCx^y.Xn), A* (f,h) « A . ( A ^ ' ( f , h ) ) f o r k = 2, 3, *i i x^ Note that A^x«£ (f,h) i s defined on G n ^ n provided that k|h| < n. n  1  A x  Now  to define the d e r i v a t i v e , we say that the function f = t(x^ '" t  has a p a r t i a l derivative  on the region G 3xJ  n  t  x) n  i f f may be altered on a  n  set of measure zero i n such a manner that the p a r t i a l derivative  aP-lf ax?-  1  exists and i s absolutely continuous with respect to x^ on any closed segment p a r a l l e l to the  axis l y i n g i n G  n  ( i . e . any segment i n G  with fixed coordinates (x^,* • • »Xi_i»x ,* • • .Xjj) ). i+1  Is thus defined uniquely up to  a set of measure zero.  on a l i n e  The p a r t i a l derivative  k  The mixed p a r t i a l  pff"  ,  pn  i s defined by i n d u c t i o n .  < l p  +  p  2  +  —  +  P  -  n  P>  Indeed, i f t h e p a r t i a l d e r i v a t i v e aP" «f p  " T - F ; — ,  Y  3x  p -i s  i—axgV  ;  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  s e t o f measure z e r o s o t h a t f o r a l l x^,' • • *x _i» s-KL»" * * » x  s  x n  *  n  a  s  8 1 1  absolutely continuous p a r t i a l d e r i v a t i v e o f order p -l w i t h respect t o x s  (on any c l o s e d segment i n G  p a r a l l e l t o the x  n  e x i s t s i n t h e sense d e s c r i b e d 9Py 3  x  l  P  l  _  " *  3  x  s  P  "  s  g  a x i s ) , the p a r t i a l d e r i v a t i v e  s  above, and we denote i t as aPs  8x7^  ^  aP-Psf  j  3x .--3x yP1  1  1  s  Note however, t h a t t h i s d e f i n i t i o n does n o t c o r r e s p o n d t o t h a t o f t h e g e 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 S o b o l e v spaces ( c f . S o b o l e v [lU]).  To v i e w 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 S o b o l e v , we  3 f show i n Chapter Seven t h a t t h e nonmixed p a r t i a l d e r i v a t i v e s3x. p  defined  I  here c o i n c i d e s w i t h t h e g e n e r a l i z e d  d e r i v a t i v e , provided  that both the  f u n c t i o n and i t s d e r i v a t i v e a r e i n t e g r a b l e on G . n  W h i l e t h i s does n o t h o l d f o r a l l mixed p a r t i a l s , i f t h e f u n c t i o n f i s i n t e g r a b l e on G , and i f t h e d e r i v a t i v e s 3 lf 3Pl^ 2f ... 3 f Q  p  p  f  9x  1 x  p  t  zx^h^  t  ax^i-'-axj!  2  0 1 1  e x i s t and a r e i n t e g r a b l e on G , t h e n t h e s e d e r i v a t i v e s a r e t h e c o r r e s p o n d i n g n  generalized  derivatives.  C o n v e r s e l y , i f t h e f u n c t i o n f i s summable on  G and has a l l t h e g e n e r a l i z e d d e r i v a t i v e s , t h e y a r e t h e d e r i v a t i v e s as den fined here.  5 F o r t h e remainder o f t h i s w o r k , u n l e s s o t h e r w i s e 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 b y r a v e c t o r ( r , " , r ;  w i t h each  ,  1  n  + p^ as above.  =  We say t h a t t h e f u n c t i o n f = f t x ^ , * * * , ^ ) b e l o n g s t o t h e c l a s s H^^ (G ,M) i f f i s d e f i n e d on G , i n t e g r a b l e t o g e t h e r w i t h i t s p a r t i a l Xl  n  derivatives  n  (k = 1, 2 , • • • , p) i n t h e p - t h power on G , and i f . f u r t h e r -  3 k f  more, f o r e v e r y 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 d e f i n e t h e c l a s s  „ (G ,M) f o r i = 2, 3, • " , n P» i  analogously.  n  x  11  I f f belongs t o a l l the classes p n H  » i^ M  X  i  f o r  l e t t i n g r = ( r ^ r g , * *' , r ) and M = (M^Mg,'• * n  11  = R,  *  =  ^» » **** * t h e n 2  n  , we say t h a t f b e l o n g s t o  t h e c l a s s Hp*(G ,M).  IfG  Hp*(M") f o r Hp(R ,M).  A l s o , we w r i t e f e Hp*(G ) i f f e Hj(G ,M) f o r some M,  n  n  n  t h e n we n o t e t h a t G  n  ^ = R  n  and w r i t e  n  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 n o t e t h a t t h e c o n d i t i o n ( l ) i n t h e d e f i n i t i o n o f H_ can be P* i V j x  ,  ,  (P)  s i m p l i f i e d i n t h e case a < 1 by r e p l a c i n g t h e second d i f f e r e n c e A ^ v f ^  ,  »' n  by t h e f i r s t d i f f e r e n c e , A„.(f . , h ) . T h i s s i m p l i f i c a t i o n i s made i n Y  c e r t a i n c i r c u m s t a n c e s t o conform w i t h a more g e n e r a l usage* f o r ease o f c a l c u l a t i o n under t h e i n t e g r a l , and i n t h e d e f i n i t i o n o f t h e norm o f t h e space Hp^Gjj).  The e q u i v a l e n c e  o f t h e two d e f i n i t i o n s w i l l be d i s c u s s e d i n  Chapter F i v e . I n t h e case G (for  = R , however, t h e e q u i v a l e n c e n  Q  o f the two d e f i n i t i o n s  a < l ) may a c t u a l l y be p r o v e d d i r e c t l y i n t h e f o l l o w i n g manner: From o u r 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 ( f , h ) • A ( A ( f , h ) ) • f ( x + 2h) - 2 f ( x +h) • fix) 2  whence we see t h a t A ( f , h ) - M(t 2%)  n-1 , , I 4 A (f,2 h)  - - |  t  2  (2)  J  w h i c h may be v e r i f i e d b y expanding t h e r i g h t hand s i d e . LEMMA 1.  I f f e L p ( R ) and 0 < o < 1, t h e n t h e f o l l o w i n g c o n d i t i o n s a r e  equivalent: i ) | | A ( f , h ) : Lp(R)|| ii) PROOF.  <  | | A ( f , h ) : Lp(R)||  c|h|  <  2  Clearly ( i ) implies  ,  a  c»|h|°.  (ii).  To show t h e c o n v e r s e , assume ( i i ) h o l d s .  11-1  A(f,2 h>: n  ^E=T  8 8  LpOOM  C (. /  1 £  Since f e I^(R)  |f(x)| dx) P  R  Il V " f:  0  R)  1 / p  + ( J  as n  |f(x • 2 h» dx) n  R  P  1 / P  ]  "* * 00  Then from ( 2 ) we have ||A(f,h) L (R)|| < ||i :  p  < £ I * j=0  1  T^gi  A ( f , 2 h ) : Lp(H)|| 2  J  ||A (f,2 h):L_(R)|| 2  J  2 J  p  l l j ^ c ^ N - c M h l ^ ^ - ^ ) 2 (2 - il) a  1  h|° - c | h | \  a  which proves that ( i i ) i m p l i e s ( i ) . Note t h a t ( i i ) may be r e p l a c e d by iii)  [ / | f ( x + h) - 2 f ( x ) + f ( x - h ) | d x ] P  S  by t r a n s l a t i o n i n t h e i n t e g r a l .  1  /  P  <— c'|h|  (3)  7 I n t h e case a = i , however, i t i s n e c e s s a r y t o r e t a i n t h e second difference, Axj^^xi** ) * 1  n  ^  m  Th  i s  >-y he seen from l i n e (3) above, o r  me  from l i n e s (20) and (21) o f Chapter F i v e . I n t h e 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 t h e f i r s t A  x  i  (f^,h) i  (when o < l ) o n l y i n Chapter S i x w h i c h d e a l s w i t h t h e norm  x  o f t h e Space. cases f o r  difference  For t h e s i m p l i c i t y o f a v o i d i n g t h e c o n s i d e r a t i o n o f s e p a r a t e (p)  9  .  a <1 and a* 1, we use A^ ( f ,h) e l s e w h e r e . i i x  x  8 CHAPTER THREE  AN EXTENSION THEOREM  The r e s u l t o f t h i s c h a p t e r i s a f a m i l i a r one w h i c h p r o v i d e s an e x r*. * t e n s i o n o f f e H-,(G -M) from G_ norm o f f •  n  to 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 t h e  Thus, i n f u t u r e c h a p t e r s we w i l l d e a l o n l y w i t h t h e 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 n o t e t h a t we p l a c e no smooth-  ness c o n d i t i o n s on t h e boundary o f G , b u t w i l l always d e a l w i t h P , n n  n  On,n  w  n  i  c  o  r  e l i m i n a t e any r e l e v a n t i r r e g u l a r i t i e s on t h e boundary.  n  We b e g i n 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 p r o o f 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  condition i n the d e f i n i t i o n  o f t h e c l a s s Hp".  LEMMA 1.  L e t f e Hp^ ^(G ,M). x  s t a n t s c - j ^ and eg n  M^'  F o r any c o n s t a n t n >_ 0, t h e r e e x i s t s c o n -  n  depending o n l y on n so t h a t t h e i n e q u a l i t y  n  Wn'Hi c i , J | f !LP(0n>M  **2  M >  n  <il  1 h o l d s f o r a l l k = 1, 2,  PROOF.  p.  F o r almost a l l ( x , " * , x ) 2  allowed i n G  n  n  t h e f u n c t i o n f may be  expanded i n t h e form  f(x +h,X2,'" , x ) = f(x ,««»,x ) + 1  n  1  n  hJL tU  lt  3x +  "' »x ) + ••• + n  (2)  x  ^ . - 4 f ( , - ' - , x ) + R(h) X l  n  p! 3x* where R( ) h  =_  1  _ /  h  (h.t)^  [  1  a  (p-l)l 0 _ f f(xi,... x ) t  3Xj  n  P  f  ^ n ^ o  t  — ^ n )  3xf 3d t  ^  .  (3)  9 This expansion i s a form of Taylor's lformula vhere the remainder (3) can be v e r i f i e d i n the same manner as the usual remainder. holds f o r almost a l l (xj_,* *'  in  G  jX^)  n > r i  /2  m  A  f o r a 1 1h  The equation (2)  ii/tn M * / n  2.  Thus we may replace h by -h i n (2) t o get  f ( x - h x , - - - , x ) = f - h - i l + ••• + (-i) iiii!£ + 3x^ p ! 3x? p  1  t  +  2  n  Isill  [/  (p-1)!  (U)  ( h - t ) * ( 3 fCxi-t X2,...,x Q _ P  1  p  t  0  n  1  3x1  pf(xi,...,x ) - — — — — — 3xJ 3  n  j dt.  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 . X o y . X n ) = (5) = 2(f +  2! 3x  ••• + — p!  2  ) + R Ch) #  3xJ  where R  #  (  h  )  =  J (p-1)!  h  ( - ) p  h  [  1  t  ^ I ^ S " - ^ )  0  "3x£  (6)  30f(xi,-«« ,x ) - 2 j — ^ 3x n  p  +  -  3 f(xi-t,x ^-- xn) , ] dt. 3x p  2  s  P  In equation (5) t r a n s f e r the remainder R (h) t o the l e f t hand side #  of the equation, and pick d i s t i n c t r e a l numbers h , Q  |h^| < ,n/2.  Then upon substitution o f the numbers  n p  ^  2  satisfying  into the new equation.^  we obtain a system of p/2 + 1 l i n e a r equations i n the variables f , 3 f 3P-P —-jjr, *"'* 3x Sx-^ 2  1  the system as  This system has a non-singular determinent, and we solve  10 32k,  0/2  - 1E  Y  i=0  3xf  i  [ f ( k K  »  1  V i» 2»'"» n X  h  X  ( k = 0, 1,  "  )  p/2),  where t h e y^ ^'s a r e numbers depending on t h e h^'s. 3  2k  P/2  f  lira 9  x  Consequently  < J  y*n.n/2>N *  :  l  I G  n,n/2  .1.  - f(x-L-b^ , x ^ • • • , x ) - R ^ h ^ ) 2  <  P  f  IY  k  ±  M  +(I G  J  1  ]| dx } P  n  l  [f(x th i ,x 2 ,---,x n ) -  1 = 0  J  I K x i + h i ^ . - " . ^ ) !  ifCx^hi^,—,x )| dx) p  n, /2  tlf  |Y | LIK  p/2  ,  i-o  ^  h  (P-D!  vS  1  - ifo 2  ±(p-1)!  l Y  f i-0  P  "  |Y  ±  k  l  J  **  ( G  n,n/2  ) I I  2 f  -",x )  3xJ  n f T l / 2  | P  | dx)  1/p  dt  +  (h.-tJ^Mlt^dt  hi  •  n,n/2  1  G  p  p  1  p  : L  /  3pf(x tt,x  ,  ^ftxi-t.xa.'".^)  f  1  *  0  n  i.*'  n,n/2)N  A  3pf(x ,'",x ) 2 — _  .  ( G  )  IR.^)!^) ^ > G  n  T  *  + ( /  1 / p  n  1*^2  5  .  0  (*) By M i n k o w s k i ' 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 [ d v ) p  1 / P  « /( / | K ( u , v ) | d v )  w h i c h w i l l , he u s e d a g a i n i n Chapter F i v e . Hardy, L i t t l e w o o d , and P o l y a , [,3l, #202;' p.  P  1 / P  du,  F o r a r e f e r e n c e see lUT.  n  v i t h t h e l a s t i n e q u a l i t y by the Holder condition i n the d e f i n i t i o n o f H (0 ,M). p , x i n» r  Thus we have 2k  where c ,n  "  3  c  ,  s  2  m  a  m  x  *  p/2 I lYiJcl k i=0 . p/2 _ 1 _ I \ k (p-l)t i=0 a  x  h  a  | /  y 1  ,  (  h i  - t ) | t | dt.  0  K  From t h i s we conclude ( l ) f o r even k s i n c e  II*  5  V  W  I  i l l *  y n,n/2>H-  !  G  Now i f k i s o d d , we suppose ( x , * * * , x ) e G ^ » 1  n  n  h  2  K  n  / 2 , and  apply the usual Taylor formula: a*"3*  3 ~ f(x,th,x , ",x ) k  1  ,  2  n  a  kH h 3 + / (h-t) 0  k + 1  Q  t  +  ~T  "' x t  T  )  l  dt.  and o b t a i n 3 " f  3 f k  k  0  Gn,n/2  i(c , l|f : L (G )|| n  p  X  L-L 2 !  * / (h-t) [ /  3  k  + h  f(x,+t,x . 3x*  Using (7)» we i n t e g r a t e ( 8 ) over G  3 f  n  » fP dx ]1/pdt  9 x lf c  • c^M) C 2  h  2  12 Thus we conclude  that  & f k  ix^ and we have completed t h e p r o o f f o r p even. (U) from (2) and p r o c e e d  I f p i s odd, we s u b t r a c t  analogously.  Now we s t a t e and p r o v e t h e e x t e n s i o n 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)| dx) P  G  < ( /  l / p  n\n  G  U(x)| dx) P  1 / P  .  n,n -+  THEOREM 1.  ^  I f the f u n c t i o n f b e l o n g s t o t h e c l a s s p ( G » M ) and n •> 0, H  n  then t h e r e e x i s t s a f u n c t i o n ty t h e e x t e n s i o n o f f from G * t  to R , n  >n  satisfying the following conditions: i ) ty i s d e f i n e d on a l l o f R , and n  II*  * L tR >ll < c j | f : V n , ) l l i n  (9)  G  n  p  n  i i ) <J> c o i n c i d e s w i t h f on G  n  n  ;  +  c  i i i ) ty e H*(M') where  P  M. i icl,nll The  constants  and c  n  depends o n l y on G  n  2 j r i  f  :  W  H  2,n i M  ;  1  = ^  2  » "'• n  ( 1 0 )  depend o n l y on n, G , and t h e r ^ ' s , w h i l e c n  and n«  B e f o r e p r o v i n g t h i s t h e o r e m , we note t h a t (.9) and ( l O ) 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 t h e Hp* norm o ftyi n terms o f the H ( c . f . (9) o f Chapter S i x . )  p  norm o f f .  13 PROOF.  F o r every p o i n t o f t h e c l o s u r e , n j * G  o  f  G  n  n*n  l  e  t  w  and :ui'  he b a l l s v i t h c e n t r e s a t 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 o f b a l l s w  l» o »  from t h e w's such t h a t  u  d  be P  P .  l t  for P e R  h ( t ) • 1 f o r t > 2.  define  n  = h ^ P ) = h( &  ±  centres  n) and s a t i s f y i n g  h ( t ) = 0 f o r t <_ 1;  h  Let t h e i r  L e t h ( t ) denote a f u n c t i o n h a v i n g continuous d e r i v a t i v e s  o f o r d e r r > r ^ (k = 1, 2 ,  :Firthermore  y u covers Xj* i=l »  )  ( i - 1, 2,  m)  ( l i )  where PP^ i s t h e p o s i t i v e d i s t a n c e from P^ t o t h e g i v e n p o i n t P i n R . n  Let H = 1 - h h * 1  2  , ,  h . m  We now make t h e c o n v e n t i o n  t h a t f = 0 o u t s i d e G , and c l a i m t h a t t h e fi  function * = fH  (12)  s a t i s f i e s t h e c o n d i t i o n s o f t h e theorem. I f t h e p o i n t P b e l o n g s t o G * , t h e n i t l i e s i n one o f t h e b a l l s  ^  >n  and t h u s 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  all P e O ^ . We see t h a t i f u <_2n/3, t h e n U ^ l C G  n u  whence i f P e OJ^ and P i s L  an a r b i t r a r y p o i n t o f t h e b o u n d a r y , L , o f G , t h e n n  PP  L  >. PIPL - P^P  Thus e v e r y p o i n t P e G  n  - G  >. n - 2n/3 = n/3.  n > u  h^ = 1 f o r a l l i , and H = 0.  l i e s o u t s i d e a l l t h e u^'s and f o r them, Then n o t o n l y i s H bounded and c o n t i n u o u s on  lit  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 also  identically  z e r o on a s t r i p i n G  o f p o s i t i v e w i d t h a d j o i n i n g t h e boundary  of G , nT  I f 0 < h < p / 2 , where p < n/3, t h e n o u t s i d e G * ^ / $(xi+h x.2*"  ' » x ) , and $ ( x h , x , * • * ,x )  t  n  1 ?  2  a  1  1 o  f  2  $  ( i » ' * * » n)» x  x  are i d e n t i c a l l y zero.  n  Furthermore  t h e r e e x i s t s a c o n s t a n t 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 o r d e r ' r > r  k  (k = 1, 2 ,  I = ( / R°  n ) are bounded above b y K.  - 2* ( P l ) Cx ,'*-,x ) + l  |#( l)(x. +h,x.."' x ) l ^ P  1  f  n  1  x  )  P  1  2  n  x  + $ l (x -h,x ,-.-,x )| dx ) l ( P  Thus f o r x ^ we have  l  /  p  n  x  (fc1') f f ^ , ( x + h , x 2 , — ^ n ) H ( ^ > ( x + h , x 2 , - - x B )  - (/ .  1  XP/2  >  =  n  1  1  - 2f ^(x ,x ,'--,x )H (  1  2  ( P  n  ^  (x ,x ,-",x )  k )  1  2  +  n  + f^\x1-h,x ,...,x )H^l- ^x1-h,x ,---,x )]| dx ) k  2  -  ( / , G  I I  n , 12  k  2  I <  1  /  P  k )  (x1+h,-",x )  n  ' ( x+ h , x , - " , x ) -  f x  =°  2  n  1  - afik)(x1,x2,--.xn) 1  P  n  f[  +  c  x  (x l  k )  -h,x  •••,x ))H 1  ( p  n  l"  n  • x / ( l-k), v (x ,x ,"-,x ) ( H ^ " ' ( x ^ h ^ , - ^ ) p  + 2f  (  v  X i  P  l "  - H ^  (  1  k )  2  n  /  x  (x ,x ,"*,x )  x^'  1  j £ ) (  2  V  h , X  n  2 " ,  , , X  *  \  )-  n  )  f  ( x ^ h . X g , * " ^ )  " xJ" H (  k ) ( x  l- » 2«'" h  X  x  , X  n  ) ) ]  P  d  x  )  l  /  P  < I < i> t / k  =°  G  * ' f Xl ( V + f  h , X  n,U/2  ,  1  |f  n,u/2  ^ (x )  x  ,  2  G  +  :  (x -h,x ,"  (  1  2»" » Si  + 2( /  |H  1 +  (Pi-k)  I'  p  t  1  "  )  l P  2  2f  (x X;L  i»V'"'xn)+  x ) r <3x )  +  n  (x x ,'",x )|  x  l f  2  x  n  1  h,x ,-,x )-H 2  ( J  (  n  |f  k ) x  x  l"  k )  (x ,x ,..-,x ; 1  2  n  (x -h,x ,-",x )| |H l- (x *h P  1  2  (  n  k )  x  G  n,8/2  .H l- (x -h,x ,...,x )| dx) (  k )  P  x  1  2  1  ]  l / P  n  Jk),  t = =0 0  l  X  v / L "» j I ^ v ^Xi » 2» G n, /2 + N  *xn^ ~  X  1  U  - 2f  (x ,X ,** ,X ) + f  (x^h.Xg,'*'^)!  #  1  2  n  |f(*)(x1,xa,-".xn)|P x  + 2h ( / G  n,p/2  1  (p.V-k+1) p 1/p x|H ^ ( x + e h , x , * " , x ) | dx ) 1  + 2h( J  |f  n,y/2  ( P  k + l )  n  (x^h.Xg,"'^)!  x  G  x|H lx^  2  (x, Bh,x ,...,x )| dx P  +  x  x  1  2  11  n  )  1  /  P  ]  db  16 <* X k=0  | ,f0^0(, x 1 + h , x 2 , " - , x n : ) , -  (I ) [ ( / 1  G n  ,u/2  X  (  l  (  - 2 f ^ ( x 1 , x 2 , — fxn) + f ^ ( x - h . x 2 , — , x n ) |  P  1  + 2h ( /  |f  (k)  * 2h ( J  |f  (k)  °n,M/2  k  P  (Xl-h,xp,"-,x ) | dx)  l/  Xl+h  i J G n,p/2 *1  =°  +  •  1/P  ]  f ( x + l ) (u,x 2 ,...,x n )du| P d x ) l / p X  l  I/"1 f l U , '"^) G 1 n,u/2 x l ~ h  t (/  + 2pl+2h  l/P  l/P  X l  O  1K  P  (x1,x?,"-,x ) | dx) xi 1 ^ n  r  dx)  {  +l)  t  2t  du|P dx) l / P ] • M j h l 0 1  +  <cl,y/2Nf;Lp(Gn)M • c 2>u/2Ml ) ), a  the last inequality following from Lemma 1, where the term Mjjhl  l  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 °n,y/2 X l  k  t / - J  f  (  n,p/2 x l "  h  1  P +2  1  • Mjhl  '2'  |f fl)(u,x2,...,xn)|P dudx^-dxj1715 ]  1  G  a  x, 1  X  +  2  h ( c 1 > y / 2 | | f : L p (G n )|| + c ^ ^ )  ).  Now change the order of integration of x.. and u with 0 .< h < \i/k: I < K ( I (j^1) 2h **. -[l"'L k=o • - * /q  \. ,ifx - Cu,X2,v-,xnH dudx 2 '"dx n ) Gn>u/U- 1  ]  17  i l l |h| + 2 a  M l  p  +  2  (c  W 2  l|f  : L (G )|| + c p  n  / 2 l ) ). M  2 f U  Again using Lemma 1 , P l  1  ~  K  -1  J n ^  (  2  h  ( C  M^hl " * 2  •  0  1  l,MA  1  M f  V  :  Gn)M +C  h (c 1 ( „ / 2 Mf  The l a s t inequality ends the computation  2.UA^  )  +  :V n ' H * ^ / A ' '  (for  (13)  for 0 < h < u/U where  we note that t h i s calculation depends on n f i r s t setting p = n/U. Now for h > u/l+ we have (p,) ( ) |« • ( x + h , x , « " , x ) - 2$ (x x x-i^ 1 P l  ( / R  L  1  ^  2  n  x  l f  ,'--,x ) +  (Pi ) n i /TI (x h,x ,---,x )| d x ) < P  r  <*||  ?  2  (J)  ^  1  1 / P  n  H  ( p l  "  k )  n  (P-I )  X  )M  : L ( < )||  Pi+2 /,} M) < 2 (c | | f : L (C )|| + p M.) K (Uh/ji) — i ,n P n ,r) x 1  n  :  ;  C  a, \  the l a s t 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 c i ^ so that equation (13) w i l l be s a t i s f i e d f o r these values o f »n 2, 1  x  n  h as w e l l . These results hold f o r a l l x^.  c  l,n *  m  a  x  c  (i) i, n  (i): c  2 n » t  =  m  a  x  i  c  2 n  *  Thus l e t  and we c o n c l u d e ( 1 0 ) . The r e m a i n i n g p a r t o f t h e Theorem f o l l o w s i m m e d i a t e l y from i f we s e t c  = maxlHJ G  n  s i n c e t h e f u n c t i o n H depends on n.  (.12)  19 CHAPTER FOUR ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE  T h i s c h a p t e r stands somewhat a p a r t from t h e remainder o f t h i s work because w h i l e t h e s e 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 b e l o n g i n g t o L ( R ) n  p  w i l l be used here s o l e l y f o r c o m p u t a t i o n a l  ease f o r p r o v i n g t h e imbedding  theorem, t h e 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. 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 g i v e s L  p  Theorem U  i n c l u s i o n i n the  e c t i o n o p p o s i t e t o t h a t w h i c h one n o r m a l l y e x p e c t s :  dir-  namely, i f g i s  an e n t i r e f u n c t i o n of t h e e x p o n e n t i a l t y p e , 1 <_ p <_ p* and g e L ( R p  then g e L , ( R ) .  F u r t h e r m o r e , C h a p t e r F i v e demonstrates an  n  p  )  t  interesting  r e l a t i o n s h i p between the c o n d i t i o n s o f 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 o f an e n t i r e  f u n c t i o n , g, b e l o n g i n g t o t h e space L ( R ) .  By t h i s we mean t h a t as a  f u n c t i o n o f n complex v a r i a b l e s z^, Zg,  Z , g i s an e n t i r e f u n c t i o n  n  p  **',  r  on C , w h i l e as a f u n c t i o n o f n r e a l v a r i a b l e s , x^, x ,  *"»  belongs t o L ( R ) . P We may c o n s i d e r t h e i n c l u s i o n i n t h e space L ( R )  of e n t i r e functions  n  2  XJJ, g  n  n  v  v  l» 2»*** n  1»?2»*  v  o f n complex v a r i a b l e s z^, z , " * » 2  r e l a t i v e t o these v a r i a b l e s .  n  ot e x p o n e n t i a l t y p e s  v  i» 2»*"* v  v n  Such a f u n c t i o n g ^ has the f o l l o w i n g p r o -  p e r t i e s : (see A h k i e z e r , [ l ] , pp. i)  z  n  lfh-119)  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  ii)  f o r any e > 0 t h e r e e x i s t s an A > 0 so t h a t f o r a l l complex  , * * *, z  n  we have  | g ( z , - " , z ) | < Ae t  1  1  n  i i i ) as a f u n c t i o n o f t h e r e a l v a r i a b l e s x-^,  x , n  g^ e L ( R ) . n  p  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 e x p o n e n t i a l  t y p e v , and f  b e l o n g s 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 sin I (_i) ( k=-~ (a-kir)^ 2  =  a  f  kir-a +  x  ) .  v  We comment t h a t o u r a c t u a l concern here i s t h a t f be bounded on t h e r e a l a x i s , and t h i s i s i n f a c t t h e case as w i l l be seen from t h e p r o o f o f t h i s lemma and as a consequence  Then we c o n s t r u c t  o f Theorem U o f t h i s C h a p t e r .  f(z) - f(0) a function g(z) = — — — — — — — z  which belongs  to  L g ( R ) , and from t h a t t h e i n t e r p o l a t i o n f o r m u l a ( l ) i s e s t a b l i s h e d . PROOF.  L e t g ( z ) be d e f i n e d 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 o f t y p e l e s s t h a n o r e q u a l t o v.  To show  t h a t g b e l o n g s t o Lv>(R) i n t h e case p > 2, we a p p l y the H S l d e r i n equality. I f 1 <_ p <_ 2 we argue as f o l l o w s .  F i r s t i f f e L - ^ R ) , then t h e  A  F o u r i e r t r a n s f o r m o f f , f , b e l o n g s t o L (R) and b y the Wiener-Paley A  A  Theorem, f has compact s u p p o r t .  But t h i s i m p l i e s t h a t f i s i n L^(R) and  thus f b e l o n g s t o L ( R ) , i . e . f i s bounded. W  S i m i l a r l y i f f e L ( R ) then 2  f e Lg(R) and again has compact s u p p o r t , whence f e L ^ R ) .  F o r t h e case A  1 < p < 2 we a p p l y the Hausdorff-Young ( l / p + 1/p* = l ) ,  Theorem t o conclude t h a t f e L^i(F.)  f has compact support and again f b e l o n g s t o Loo(R).  In  a l l cases f i s seen t o be bounded, whence we conclude t h a t  2  /  |g(x)| dx < «.  Then on t h e b a s i s o f t h e Wiener-Paley Theorem a g a i n , t h e r e e x i s t s a f u n c t i o n * e L ( - v , v ) f o r which 2  / \  1  g(z; = — 2TT  rJ V  i z u  e -v  /\  *(u)du  or z  f(z) = f(0) + — 2TT  or i n s t i l l  another  v  form,  f(z) = f(0) - —  2ir  We  , izu / e $(u)du -v  J $()i-(e )du. V  iZU  u  -v  8u  are i n t e r e s t e d i n the e x p r e s s i o n  U(x) = sinct f ' ( x ) - vcos a f ( x )  22 w h i c h may be expanded i n t h e form U(x) = -vcos a f(0) + 1 + —  , V  2ir  g  , / *(u) — -v 3u  We now decomtiose t h e  ( e  ( u s i n a + i v c o s a) ) du  function  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 w h i c h t a k e s t h e form V(u) = v s i n a \ (-1) k=-°°  g e (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 a p p l y i n g  (3) we get ivsin^ct U(x) = -vcos a f(0) +  x  2ir  x Jr »(, u )—  ,?  V  -v  (  (I  3ft -  "  l ) k  e  2  00  (a-kir)  i u ( ( k r a ) / v + x). )du  2 = -vcos a f(0) - v s i n a «  x  J -  x  .(-I)' ) -r? 2 ( f ( k i r - a / v +x) - f(0)) (a-kir) k  00  2  £  = vsin a £ f ( k i r - a / v +x) -» (a-kir) 2  with the l a s t i n e q u a l i t y holding 00  1  ( r l )  ka=-<» (a-kir)  d  k  5"  since 1  =  c o s  =  da s i n a  a  ~ * sin a  This proves ( l ) . For a complete d i s c u s s i o n o f 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 m u l a , we r e f e r t o A h k i e z e r , [ l ] , pp.  182-189.  W i e n e r - P a l e y Theorem we r e f e r t o Zygmund,  [l8],  t h e Hausdorff-Young Theorem,  As a source f o r t h e  [l8],  pp.  272-27^, and f o r  25U-258.  pp.  The theorems o f t h i s c h a p t e r a r e p r i m a r i l y b a s e d on t h e f o l l o w i n g lemma w h i c h i s an immediate  consequence  o f Lemma 1.  LEMMA 2. I f t h e f u n c t i o n $ ( x 2 , " * , x ) i s , r e l a t i v e t o x-^, an e n t i r e n  f u n c t i o n o f t y p e v f o r almost a l l ( x p , " * , x ) and <J> e L ( F r ) , t h e n ix  P  belongs t o  ±  and 3* II—: dx  PROOF.  L (R )  || v | |  n  £  * : L (R )||. n  p  p  ±  As a f u n c t i o n o f x^, $ s a t i s f i e s t h e c o 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 o b t a i n 3x  ±  = -nir  I -i" ^ $ U ( k - — ) / v +x, , x , " * , x ) . k=-» ( k - ? l / 2 r 1  9  9  n  Thus ||^-: 3x  L(R )||  <vjf _ i _ | | k=-» ( I - 1/27*  n  *  2  p  x  L (R )|| n  :  p  = v|| * : L ( R ) | | n  p  V since  l k=-»  1  *  (k-  1/27 ir  - 1.  T h i s completes t h e p r o o f .  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 function o f types v^,  v , and g^. e n  L ( R ) , then n  p  IhS 3^  : L ( R ) | | < v ||g. : L (R )||. k p n  n  P  v  The f o l l o w i n g t h r e e theorems a r e r o u t i n e , b u t w i l l l e a d t o t h e 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.  L e t g ( z ) b e l o n g t o L ( R ) and be an e n t i r e f u n c t i o n o f e x v  p  p o n e n t i a l t y p e v ; l e t a > 0, h = a/v, x let x  satisfy  k  x  i  <_ x' <_ x ^ + r h where r i s an i n t e g e r . k  k  pa/p  PROOF.  (1  r«)||g  +  Then  : L (R )||. n  v  Clearly  t)  k+1 |g (x)| dx*I J _» ~~ k 00  /  <  I |g ( x ^ l P ) i / P  ( h  = k h (k = 0, £L, +2, • • • ) , and  k  00  » |g ( x ) | d x = h l  X  P  p  v  |g  r, (t)|  P  x  where x ^ <_ 5  k  ^ x^+i*  we have b y H o l d e r ' s  »  1 / p  Furthermore, u s i n g Lemma 2 and s e t t i n g 1/p + 1/q  inequality  k(<)i  i<!  p  igv(?k)ip)1/pi  )^.(!  k=-m  k=-m  <» cj  - <VlV  1/p  gv  /p  k=—m 1/p  m  x  k+l  .  - ,  p 1/p  <V t i (/ !g;(t)idt)} x^ , . _ x,,+rh p p/H lh [ I j |gv(t)|dt (rh) ] k=-m  1/p.  m  i '/ \ i P  K  t  k=-m  ,  xP/<l,l/P  ^  P  1/p  <h(r f |g'(t)|dt) 1+P/q  * rh||g^  : L (R)|| < p  ra||gv  : Lp(R)||.  25 Therefore h  1  /  p  m < i ig(x;)i ' k=-m p  1 / p  K  v.1/p  r ? I / *\ Ip r , , t %|Pil/P I i | g ( x ^ ) r - 1 |g v (5 k )n k=-m k=-m  = h  < (1 + r o ) | | g L e t t i n g m -»• THEOREM 2.  ve obtain  T  1  /  P  : L (R )||. n  v  (5) and t h e p r o o f i s complete.  L e t g-*- b e l o n g t o t h e space L ( R ) and be an e n t i r e f u n c t i o n n  P  V  of types v  ? I , MPX •vl/p, h (k<=-m 1 lg/vl )  A  l f  v ; let  < 0, h  n  k  = c ^ A ^ (k = l,*«« n), x t  k  = sh  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 t h e i n e q u a l i t y sup [ ( I k ) u k = l * -~ A  I |fo(x* - u , . — - u — l n v  s  1  s  )| ] P  1  /  P  n  (6)  n <. H ( 1 + r a )||g* : L (R ) | | , k=l x v p where t h e sup extends t o a l l r e a l u , ***» ±  and t h e summation i s  c a r r i e d out on t h e i n t e g e r s s^,»**, s . n  The p r o o f o f Theorem 2 i s b y i n d u c t i o n from Theorem 1. the  We omit  details.  THEOREM 3.  L e t g^ be an e n t i r e f u n c t i o n o f t y p e s v »***» v x  n  f o r which t h e  _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 t h e c o n d i t i o n s k k s o f Theorem 2. Then g^. b e l o n g s t o L ( R ) . and °^ p 1 1  CT  Mg-  :L  (R )|| ! [ ( 2 r ) N  p(  I  h.]  x  1 / P  J — Ig^x" -u^—.x* -*° -°° "nS_ CD 00 V S il  x 8U  u. 11  PROOF.  S  n  1  v  P 1/P  -u n )( )  s  We must show t h a t ||g+ : L^CR ) | | i s f i n i t e . 11  eo  oo  J •••/ | g ^ ( x , * " , x ) | d x « " d x P  1  n  1  oo T l  = I'" I -°°  -  / J  ^ 1  ,  - J J  -1  X  0 0  » 2rh  -»  </  2rh  n  2rh  •••/ o  0  -1  s  -» 0  2rh  |g^ 1 ,-,x n )|V-dx n  N  X  Sl  00  n  ,  0  1  00  i  00  |g*(x"  I'" I -»  -»  v  J—  < ( 2 r ) ( ni^) sup n  -°°  k=l  _  I _oo  s  n l 1  u s  n  p u )| dun'-'du n  1  |g*(xt - V " , * ^ 1  n  -uDp.  n  T h e r e f o r e | | g ^ : L p ( R ) | | i s f i n i t e , and we conclude ( 7 ) a f t e r r a i s i n g n  t o t h e power 1/p. Tl/v  We note t h a t t h e c o n s t a n t ( 2 r ) following variation:  may be  o m i t t e d i n (7) w i t h t h e  i f the points o f interpolation x s  s p a c e d , t h a t i s i f we r e p l a c e x S  h  f i c i e n t t o take the i n t e g r a l  J 0  i n the t h i r d l i n e o f the proof.  by x k i  s  are e q u a l l y  i n ( 7 ) , t h e n i t w o u l d be s u f k 2  r  f o r approximation i n s t e a d o f J 0 T h i s would e l i m i n a t e t h e c o n s t a n t  n/p (2r)  k  , and i n e q u a l i t y (7) hecomes  h  i  Me* : V  Rn)!l  < (m,/'* s u ( f . . . ll^x  .  1  P  -»  -«»  U l  ,-.,x  1  -u )|  n  )  p  n  1 / P  .  n  E x p l i c i t mention may now be made o f t h e 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 d e f i n e  ((g+ : L„(R ) ) ) n  = sup[(n h ) k  00 00 X-..  -»  Uj  Jlg^Cx  -»  -u )f )  1  1  /  p  n  l  s  n  as i n Theorem 2, we t h e n have t h e i n e q u a l i t y  ||gg LEMMA 3. a  l»  : L (R )|| < n  p  C(g*  : L ( R ) )) < n  p  n(l +  c^Ng*  rL (R )||. n  p  F6r 1 ^ p <_ p' < » and f o r a l l n o n - n e g a t i v e numbers  * * *» '"n  w  e  n  a  v  e  (X \ k=l  )  (X \ > •  i  K  k=l  K  U s i n g Theorems 1,2,3 and Lemmas 1,2 we now prove t h e s i g n i f i c a n t r e s u l t s o f t h i s Chapter w h i c h g i v e ( f o r e n i t r e f u n c t i o n s o f t h e expone n t i a l t y p e ) L (]R ) i n c l u s i o n f o r d i f f e r e n t v a l u e s o f p and d i f f e r e n t n  dimensions o f R . 11  These two theorems w i l l be used t o prove t h e m a j o r  r e s u l t o f t h i s work w h i c h i s t h e imbedding theorem o f Chapter 7. THEOREM h. types  I f the function S ^  i s  an e n t i r e f u n c t i o n o f e x p o n e n t i a l  y., ***, u and b e l o n g s t o L ( R ) , and i f 1 <_p < p" <_» , t h e n n  g ^ b e l o n g s t o L i ( R ) , and n  p  28  18+ *V PROOF.  (Rn  (  2  V*)1/P "  1/P  >ll l " k  (10)  ' Ng- : V R n ) U-  From t h e v a l i d i t y o f (8) and Lemma 3,  < (; k=l  u  n  -Un)ip'  i — i  (  i  -°°  -«  co  oo  -i /_»  1  s  > l/p '  n 1/r)  n  < (n ^r sup ( i — i \^(x] -n ~\* k=l u^ -» -» l n n l/t) - l / n ' < ( H h.) n (1 a j | | f o t : L ( R ) | | k=l ^ k=l ^ 1 5  )  n  lt  v  s  s  n  n  +  v  =  n (1 • o^) (v \) k=l  The f u n c t i o n ( l + a)/a  w  U)» CO  : L (R )|| n  p  n  Let u = l / p - l / p ' .  a t t a i n s i t s miniraun (on t h e i n t e r v a l (0,l) ) o f  xl  (1 -  Therefore i t follows  ||g- : L ( R ) | | ,  > 0 (k = 1, 2, *•*, n ) .  w h i c h h o l d s f o r any  l|g+  " ^  1/V  k  P  -  W  (11)  <— 2.  to)  that <2 n  n i fn ( n v ) k  1 lr>' \\  : L (R )|| n  H  p  <».  Then g-> e L i ( R ) and we have completed t h e p r o o f , v p n  THEOREM 5.  I f g ^ i s an e n t i r e f u n c t i o n o f t h e e x p o n e n t i a l  type b e l o n g i n g  t o Li ( R ) and i f 1 < m < n , t h e n f o r a l l f i x e d x , , **•, x is P — m+l' * n v n  B  29 i n t e g r a b l e i n t h e p - t h power, and n( n  2  lie*:  *t ly\  V  ^ °  y^w  N « $  :  L  P  (  R  N  ) H -  (  I 2  ^  m+1 PROOF.  From ( 8 ) we have t h a t  n (1 + ^Hg^ k=l > (n h )  l / P  k  : L (R )|| n  v  s u p [ I .-I " i  1  > (H h^) 1  > (n h , )  1/p  s  ujs  1  ^  aftl  m + 1 /  h )  1 / p  k  1  sup ..,s  sup [ V ' . U m  £(n h k ) 1 / P [ ( n  Ig^x* - u ^ ' - . x "  l n  s  .«p  1  which holds  y  I s  8  I  for a l l a  1  lt  s  l  s  s ffi  l  1  s  l  s  m  -u )| ] P  n  1 / p  n  s  • ^ ^  1  . - ^ ) |  ^ . — j?  V  K  n  ( nh k ) l / P ||gj m+1  and we conclude (12) n o t i n g t h a t h ^ = k / k ' a  v  1 / p  P  m  1  function o fm variables:  p  l  -u ,-**,x )| )  > 0. We l e t a. = 1 and a p p l y ( 8 ) t o g-> as a  : L (P )|| >  p  m  ...I ! g ^ ( x j  sup  k  2n|.|g*  1 / p  n  : f g ^ -u,,".^  m  V " * ^  1  ' " I Ig^x -n "\x  - I i  p  n  n  [ n  -« )| ]  : L (R * m  p  n  1 / p  ]  30 CHAPTER FIVE  APPROXIMATION OF CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS OF THE EXPONENTIAL TYPE  We now b e g i n t h e t e c h n i q u e o f q p p r o x i m a t i o n o f c o n t i n u o u 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 .  functions  W h i l e t h e r e a r e o t h e r 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 t h e n e x t r a p o l a t i n g b y " d e n s i t y " )  this  a p p r o x i m a t i o n a l l o w s us t o c a l c u l a t e t h e " h a r d c o r e " i n e q u a l i t i e s using t h e convient  e n t i r e f u n c t i o n s r a t h e r t h a n t h e c o n t i n u o u s ones.  Let f e LptR ). 11  also belonging  We may c o n s i d e r a l l p o s s i b l e f u n c t i o n s g ^ x ^ , * • • , x ) n  t o L (R ) and h a v i n g t h e p r o p e r t y t h a t f o r almost a l l 11  (x2»"*,x ) t h e y are e n t i r e f u n c t i o n s o f e x p o n e n t i a l type v w i t h n  t o x^.  respect  As a measure o f how c l o s e l y t h e 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 , we d e f i n e t h e q u a n t i t y : v  where t h e infimum I s extended o v e r a l l p o s s i b l e g . y  LEMMA 1. 3 f — ax^  I f t h e f u n c t i o n f b e l o n g s t o L ( R ) and f has p a r t i a l d e r i v a t i v e s n  p  P  also belonging  P  ||A  PROOF.  t o L (R ) , t h e n f o r a l l i n t e g r a l s >^0,  (  + 8 ) x  (f,h) :  Lp(R )I| n  < h ||A P  S  (f  ( P )  ,h)  : L (R )||.  I t s u f f i c e s t o show (2) f o r p = 1 and s = 0.  n  (2>  |A  ( f , h ) : L ( R ) | | = (/-«/  |A  n  eo  P  1  x +h — / I/ *i (t,x ,-",x )dt| dx ...d -» x 1 1  09  /  P  2  -  1 / p  n  oo  - (/  n  1  X n  )  1 / P  x  < (/" -~f\t  | dt h P  x  —oo  <h  (f,h)| dx ...dx )  d x ^ - d x / ^  P / < 3  _oo 1  l / p + 1/q . . °° . i ,p .l/p (/ "'J | f | dx ••*dx ) —CO _oo l r  n  x  -h I If  : VR )M n  x  l  p  where l / p + 1/q = 1. ||A  (  Thus  ( f , h ) : L ( R ) | | i h M A ^ - ^ C f * ,h) : L ( R ) 1 l l + S )  X  n  n  p  < —  x  x  p  , P i i , s , (p) < h ||A ( f : L (R )||. x x^ p P  S  V  P  /  n  ±  We now e s t a b l i s h t h e 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 t h e approximation 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 p r o o f 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 w h i c h approximates f t o w i t h i n a s p e c i f i e d distance. THEOREM 1.  F o r a l l r > 0 and a l l v , t h e r e e x i s t s an e n t i r e f u n c t i o n  (r) K  (u) o f exponential type v s a t i s f y i n g t h e f o l l o w i n g three i ) /" K  ii)  ( r )  (u)du = 1  f " | K ( u ) | i . d u < a„ \\ • r (r)  conditions:  iii)  f o r a l l f u n c t i o n s f "belonging t o H  (M) we have t h a t P» l r  x  oo r \ | [f - J  K ^ (u- )f(u,x ,"»,x )du : L (R ) j | <^ b ^ / v —00 P 11  Xl  where t h e c o n s t a n t s a , b r  r  2  n  depend on r b u t n o t on v.  From (5) we con-  clude that A for a l l f e H  x  Cf) p <-  b_M/ r  v  (M).  r  P.x PROOF.  v,x  x  L e t g ( z ) he an e n t i r e f u n c t i o n o f t y p e u n i t y s a t i s f y i n g t h e  conditions 00  i) /  g ( t ) d t = 1,  _oo oo ii) J  j g ( t ) l d t < »,  oo iii) b  r  = /  |g(t)|  | t | d t < oo. F  —CO  We f i r s t n o t e ,  v  p+l  (-if  *  that  P+2, , P+ , i-l 'A (*,h) = I (-1) i=0 2  x  ,p+2, , ( . )*(x + i h ) P  1  I p. * ( x + i h ) - $ ( x ) i=l  , . i - 1 ,p+2. where p. = (-1) ( . ) . Then 1  Now l e t f e H  P.*!  p  t  2  £ p i=l  =1.  1  (M). C o n s i d e r t h e f u n c t i o n d e f i n e d b y  33  g ( v  X l  ,--\x ) n  _» 00  = /  —  g(t)[(-l)P+14 00  P + 2 )  (f,t/v)+f( ,---,x )]dt X l  1  n  p+2  «•  = /  f  -  g(t) I P ^ ^ i=l  + t/v,x ,'-',x )dt 2  (r) K (t - x ) f ( t , x , " ' , x ) d t 1  2  n  where  P+2  ( \  '  K  =  A  P i V / 1 g(vu/i)  *  (r) Then K  i s an e n t i r e f u n c t i o n  y  r e s p e c t t o x^. g  o f t y p e v , and hence so i s g^ w i t h  Moreover,  - f = (-1)  J  P +  g(t) A  P + 2  -oo  1  (f,t/v)dt,  -»  Pr  Thus from (2) and from f ' s membership i n H  1  we have  r  P »i x  ||g  - f : L (R )||  <  n  M/v  P+Ct  "  /  |g(t)|  |t|  P + a  d t = b M/v  —oo  w h i c h p r o v e s ( 5 ) . From ( 7 ) , ( 1 0 ) , and ( l l ) we c o n c l u d e ( 3 ) .  oo  J  ,  p+2  \  |K* '(u)|du< r  —oo  =  I  oo  |p | J  1  V  P+2 I  v / i |g(vu/i)|du —oo  oo  IPJ /  r  r  |g(u)|du  and t h e p r o o f i s c o m p l e t e .  = a , r  Furthermore  (12)  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 t h e a p p r o x i m a t i o n o f functions belonging  t o t h e c l a s s H (M) b y e n t i r e f u n c t i o n s . p  that the quantity A+(f)  =  •••  A  p  v  ^ ^p f  v  S^^  i s  11  We n o t e  ^  Anj(f) = i n f | | f - g-+ : L ( R ) | | P g v p  (13)  n  v  v  where t h e infijBum i s t a k e n o v e r a l l e n t i r e f u n c t i o n s g ^ o f e x p o n e n t i a l  types Vjj. i n the variables x^, k = l t * * * , n . THEOREM 2 .  I f t h e f u n c t i o n f b e l o n g s t o t h e c l a s s H (M"), t h e n p  *  k = l v. k  K  where d depends o n l y on r , ***, r . ±  R  We g i v e an o u t l i n e o f t h e p r o o f f o r g e n e r a l n.  For a f u l l  dis-  c u s s i o n o f t h e case n = 3 we r e f e r t o [ 9 ] , pp. 2 5 9 - 2 6 0 o r pp. 1 7 - 1 9 i n the t r a n s l a t i o n .  (r ) ±  PROOF.  Let K  be d e f i n e d as i n ( l l ) f o r i = 1 , ,  ( r  FT  l>  ( r  n  n.  }  n  1  x f t x ^ u ^ X g + U g , * " .Xjj+Ujjjdu, and (r )  ( r )  2  R  n  I  n - f (x,+u, ,x +u ,• • * ,x +u ) ]du = l\\ k=l 2  2  K  where t h e ifi^'s a r e d e f i n e d as f o l l o w s :  Then  35  (r.) *1  / n R  =  (r)  v  K  <V  n  v  1  t t t x ^ r " , * * )  f(x +u ,x V* ,Xn) ^ ,  1  -  n  1  d  u  2  - (r.) = /  *  2  v  K  ( u  i  )  f-(  [  - / K^^u^'-'K R*^ 1  y , x ) - f t x ^ . x g / " ^ ) ] du  X ; i  n  (u ) [ f t x ^ . X g , — ^ )  n v  n  x  -  n  f ( x + u ,x +U2 , x , ' *' , x ) ] du x  x  2  3  n  (r,)  00  = J  K  (u ) h (x +u ,x ,-",x )du  v  x  2  (r.)  »i -  v  x  1  1  K^"*  (u  v  n  n)  ,  1  1  i  —/  -co  i + 1  [ f ( x  K  _»  1 V  l  n  1  , , ,  ' U i-l' l» ' n x  + n  x  ] du  (U^) x  1  i-1  ^  x  , • * * ,x ) du * * • d u j . ^ n  x  where  h  i < l i•*''. i-l i-l» i • * *'» n x  +u  x  +u  x  x  }  (rj = K . (u.) [ f C x ^ . - " ^ ^ ^ , - - - ^ ) v  - f(x +u ,*",x +u ,x x  1  , M  (r, )  (V-K/-  x h (x +u , • • • x  l+ttl»  ,"',x )  (r,)  00  ±  n  (r)  - f(x +u ," ,x.+u ,x  ~ J  2  i  i  i+1  ,***,x ) ] . n  x  i  "  Then b y (5) h  :L  n  M i p^ )M l ^ V ^ and II*!  Rn  V )H < K*!^*  :  1  »  n  i i*i *y* >n <f ' i V ^ v - ^ ^ V . ,)i * _eo  *  x  (  -«>  i - 1  1  / lhi(V^»*,^xi-l+Vl»xi»'''»xnHPto)1/PdV,'d^ n  < a r  a  l  ---a ||h i - l  : L (R )|| n  r  2  r  < a  1  p  (b^M./v. " ), 1  ( i = 2,  1  n )  F i n a l l y we c o n c l u d e t h a t  f - f o : L (P )|| < n  b  M  T/  b  x  1  r  i »  l  r  n I IU k=l  M  : L (R )|| n  k  v  b M  2  v / 2  r  r  n  d M  r n  n  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 w i t h o u t p r o o f . p r o o f may be found i n  [ 9 ] ,  pp.  2 6 1 - 2 6 2 ,  o r pp.  1 9 - 2 1  A  i n the t r a n s l a t i o n .  37  LEMMA 2.  L e t ( f ^ ) he a sequence o f f u n c t i o n s each o f w h i c h b e l o n g s  t o L ( R ) and has p a r t i a l d e r i v a t i v e s n  also belonging t o L (R ). 11  p  If ||f - f  : L (R )|| * 0  «sk + -  n  k  and  p  p  3 f a : L ( R " ) | | •»• 0 P p 3x, l  $  as k f »  v  t h e n f may be m o d i f i e d on a s e t o f measure z e r o s o t h a t t h e f u n c t i o n 3 f ~TT P  f has t h e p a r t i a l d e r i v a t i v e  satisfying  dx^  9 f P  — 3  = x  l  We now e s t a b l i s h a c o n v e r s e 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 w h i c h can be s u i t a b l y approximated b y e n t i r e f u n c t i o n s r a c t u a l l y b e l o n g s t o t h e space H THEOREM 3 .  p  L e t r >. 0 , f e L p ( R ) and suppose t h a t n  A "  (16)  1 K/\>  r  k  for a l l  v  ( k = 0 , 1, * * * )  running through a geometric progression a  w i t h commom r a t i o a  >  1.  /Fjurthermore, l e t g d e s i g n a t e an e n t i r e  f u n c t i o n , w i t h respect x^ o f e x p o n e n t i a l type u n i t y f o r which  Wt - g : L (R )ll I K .  (17)  n  p  Then t h e f u n c t i o n f , = f - g b e l o n g s t o t h e c l a s s H * 1  M - CjJC w i t h c  r  depending on r b u t n o t on K.  (M) where  PROOF.  Let g  in  v  = g (x '* v  * , x ) d e s i g n a t e an e n t i r e f u n c t i o n o f t y p e  lt  R  f o r which ||f - g  : yR  v  1 1  ) ! ! < K/v  J  where we make t h e c o n v e n t i o n t h a t v = a  = m  k  0, 1, • • • ) .  (k =  k  Then because f s a t i s f i e s t h e c o n d i t i o n s  o f t h e theorem, i t may be r e  p r e s e n t e d i n t h e form f • g + * where g = g  and  m  o »  *"k k - l  k=l  m  (We note t h a t t h i s t e c h n i q u e w i l l be used f r e q u e n t l y i n t h e s e q u e l . ) 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 t i m e s w i t h r e s p e c t t o x .  Then  ±  l  x  k=l  where (gm " Sm _ )  9  k  \  k  " 9  x  ±  F i  From Lemma 2 o f Chapter U and (16)  r+1 n ||Q  :  k  y R  P )||  <  m^  2K —  kp =  a  ^k—1  2K a  (  K  J  ) (  P  a +  « J  1  a  ! ~  —ka Because t h e s e r i e s dominated b y a v e r g e s i n p ( ^ ) norm. L  n  c o n v e r g e s t h e s e r i e s (19) c o n -  Thus b y t h e p r e c e e d i n g Lemma, t h e f u n c t i o n  $ b e l o n g s t o L (R ) and has a d e r i v a t i v e o f o r d e r p , and t h e s e r i e s 11  (19) converges t o t h a t d e r i v a t i v e i n L ( R ) norm. n  39 Now l e t h > 0 and f i n d a n a t u r a l number N f o r which . 1/a  N  +  1  <  h  n <_ l / a , w  or e q u i v a l e n t l y  a  N+1  >  1/h  >  K  a  Then we have t h a t  |A* ( * \ h ) : L ( B ) | | < l l ( p  x  n  x  P  N k  =  2  1  1  P  k=N+l  K  P  Now from Lemma 2 o f Chapter U and i n e q u a l i t y ( 2 ) o f Lemma 1, we have that N  2  II|A 1  X  (Q ,h) : L ( R ) | | 1 k  ± h  p  2 r , (g) lrt  _ , n. , .  : L(R )|| » l P  1  n  (2-a)(N+1) r+1 '  a  2 5 2k  < h  x  2  a  1 0 1  2  ^ 1  a  2-a ~  a  2 = 5 — 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  ka  N+1 a  < > 20  Uo From t h i s we c o n c l u d e t h e theorem b y s e t t i n g r+h  c  r  r+3 « a  +  * i — a -1  , . (22)  .  We make s p e c i a l n o t i c e h e r e o f t h e e q u i v a l e n c e o f t h e spaces d e f i n e d b y t h e 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 t h e p - t h d e r i v ative of f. for  a < 1.  Theorem 1 may be p r o v e d u s i n g o n l y t h e f i r s t  difference  That i s , a f u n c t i o n s a t i s f y i n g t h e H o l d e r c o n d i t i o n on t h e  f i r s t d i f f e r e n c e o f t h e d e r i v a t i v e may be s u i t a b l y approximated b y entire functions.  The c r u c i a l n e c e s s i t y f o r s e p a r a t i o n  o f t h e cases  ct < 1 and a = 1 i s demonstrated i n t h e p r o o f o f Theorem 3 where t h e i n e q u a l i t i e s (20) remain v a l i d u s i n g t h e f i r s t d i f f e r e n c e f o r «< 1, b u t f a i l t o h o l d on t h e 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 functions w i l l s a t i s f y both the first the  and second d i f f e r e n c e s on i t s p - t h d e r i v a t i v e i f a < 1, b u t o n l y  second i f a = 1. T h e r e f o r e when, i n t h e f o l l o w i n g theorem, we n o t e  t h a t t h i s a p p r o x i m a t i o n i s b o t h a n e c e s s a r y and s u f f i c i e n t  condition  f o r membership i n t h e H - c l a s s , we w i l l have demonstrated t h e e q u i v a l e n c e for  a < 1 of using the f i r s t  o r second d i f f e r e n c e , w h i l e i t remains  n e c e s s a r y t o r e t a i n t h e 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 t h e f u n c t i o n g o f expone n t i a l type u n i t y w i t h respect  to x  1  a l s o belongs t o the c l a s s H  b u t o n l y w i t h a d i f f e r e n t c o n s t a n t depending on | | f : L ( R ) | | . n  r Pt l x  In  t h i s case i t f o l l o w s from ( I T ) t h a t ||g : L ( R ) | | < | | f : I ( R ) | | + K n  p  n  p  (23)  h  and from Lemma 2 o f Chapter  * VH )M  Us^  t  < | |f : L (R )||  n  n  p  +  K  .  Thus  M^Cs^.h)  : L (R )|| n  p  < k ( | | f : L ( R ) | | + K) n  p  and from Lemma 1 o f t h i s c h a p t e r we have t h a t  MA*. ( g \ h ) : L ( R ) | | {  n  x  x  l  x  <  l  p  h||g  ( p + l )  l  x  The  : L (R )|| < h(||f : L_(R )|| " n  n  +  K  ) .  l a s t two i n e q u a l i t i e s i m p l y t h a t f o r a l l h  Combining t h i s w i t h Theorem 5 o f Chapter  h and Theorem 1 o f t h i s  Chapter  we have t h e p r i n c i p a l r e s u l t o f t h i s Chapter which w i l l be t h e b a s i s o f all  future calculations.  THEOREM k. H*" _  P»  In o r d e r t h a t t h e f u n c t i o n f e L p ( R ) b e l o n g t o t h e c l a s s n  ( i . e . H**  (M) f o r some c o n s t a n t M) i t i s n e c e s s a r y and s u f f i c i e n t  P * 2_  x  x  t h a t t h e r e e x i s t s a constant K f o r which  V  (  f  ,  p  f o r a l l v >1, o r at l e a s t le gression v = a The H  .  r  P.  x x  i  I  /  ,  r  f o r a l l v running through  a geometric  pro-  ( a > l , k = 0 , l , **•).  sufficient  c o n d i t i o n f o l l o w s from Theorem 3 by a d d i t i v i t y i n  F i n a l l y we f o r m u l a t e r e s u l t s e q u i v a l e n t  t o Theorem 3 f o r t h e  Hr  class  p  V n  n. R ) satisfy  K, i  where K^'s a r e c o n s t a n t s , progression  » a^  r ^ > 0, and t h e v^'s r u n t h r o u g h a g e o m e t r i c  (k = 0, 1, *** ) w i t h a.. > 0.  L e t g he an e n t i r e  f u n c t i o n o f t y p e u n i t y i n each o f t h e v a r i a b l e s n | | f - g : L (R ) | | <  such t h a t  n  I K..  1=1 - + Then t h e f u n c t i o n $ = f - g b e l o n g s t o t h e c l a s s H (M) where P  P  n  i  X  M. a" Cr u  T*i_.1=1  1  K  i*  J  J  PROOF!.  R>r  the approximation o f type  with respect t o x  1  we  obviously  have t h e i n e q u a l i t y n  K.  L. ( f ) < A*(f)_ < J — r r x  v »  P -  x  1  1  f o r any v _ , * * *, v . <£ n  v  Thus K  V,v  (  f  ) P  i * i  rx  p  i  v  l  ^  — i - 1 7 i i  I t f o l l o w s t h e n from (27)  v  and (26)  t h a t , on t h e b a s i s o f Theorem  3,  t h e f u n c t i o n $ as a f u n c t i o n o f t h e v a r i a b l e x-^ b e l o n g s t o t h e c l a s s r  H  l  P.X-L  (M. ) where 1 n 1  r  l 1=1  1  The same argument h o l d s f o r X 2 » " " » x n » m&  t h e p r o o f i s complete.  1»3 CHAPTER S I X THE NORMED SPACE H^ p  W i t h t h e t e c h n i q u e o f a p p r o x i m a t i o n "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 o f c l a s s e s o f f u n c t i o n s d e f i n e d n R .  over  r I n t h i s c h a p t e r we prove completeness o f t h e 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 c h a p t e r we use t h e e q u i v a l e n t  definition  ? r f o r H • namely, a f u n c t i o n f b e l o n g s t o H (M) i f f has d e r i v a t i v e s P P» j x  £ a l l b e l o n g i n g t o L ( R ) f o r k = 0, 1, " * , o p  ||A  (f  v  \\L 2 x  i  (f  and i f  ( P )  ,h)  : L ( R ) | | < M(h|°  (a <l )  ( p )  ,h) i  : L ( R ) | | < M|h|  (a « l ) .  x  n  n  p  The f o l l o w i n g two lemmas l e a d t o t h e i n t e r e s t i n g Theorem 1 r w h i c h , i n t u r n , i s used t o prove completeness o f Hp. LEMMA 1. iable x  k  An e n t i r e f u n c t i o n g = g + ( x , * * * JXJJ) o f t y p e s x  which  satisfies ||g  :LfR )|| < M n  b e l o n g s t o t h e c l a s s H (M ) where ' M PROOF.  x  , i = UMVj . r  From Lemma 2 o f Chapter U and Lemma 1 o f Chapter 5,  i n the var-  I*x ^ . h ) : L ( R ) | | n  if  [ v h | <_ 1.  On t h e o t h e r hand, i f  ±  " * A  (  « x <  i  x  '  i >  h )  V "i  :  Rn)  i  X  P i  a i  concludes t h e p r o o f f o r  MA^g^.h) : L (H )|| n  p  whereas  ( p  |Vjhj  2  i + l )  :  L (R )|| n  |h|  1,  >  n  M * ( P i ) : V* >N  P  2Mv |v.h|  This  ||g ;  <  P  X j  = 2MvT |h| . i  ai  < 1.  I fa  <«vJ  Pl+2)  x  = 1 and i f |y.h| <_1  |h| <Mv^|b| , 2  i f |v h| > 1 i  " x A  ( g ( 1  xl ' >  h )  V >M<Ml f  5  pn  ( g x  : L (H )|| n  p  Pi r* <_ UMVi f.'tMVj |h|.  LEMMA 2. the  Let the function  f = f(x,,**',x„) be t h e sum i n L -norm o f i* *n p  series 00  fUi.-'-.Xn)  «  I  Qs(xi,**-,x ) n  s=0 s/rjj where Q  g  (s=0, 1, •••) i s an e n t i r e  in the variables  x  k  function  o f types v  k  = 2  which f u l f i l l s t h e i n e q u a l i t i e s  llQo '  V !! ± Rn)  A  M°s V*"*!! <.B/2 (s = 1, 2, ••• ) :  8  where A, B , and a a r e a r b i t r a r y , not depending on s. Then f b e l o n g s to the class H  (K) where r  x  = ar^ and %  = *** = K  n  = K<_UA + B.  Furthermore, | | f : L ( R ) | | <_ A + cB n  p  where t h e c o n s t a n t c does n o t depend on A o r B. PROOF. L e t  u-1 3=0  By t h e c o n d i t i o n s exponential  o f t h e theorem, S^- i s an e n t i r e f u n c t i o n w h i c h i s o f  type l e s s than u/r> v  = 2  fc  i n t h e v a r i a b l e x^.  ( u = 1, 2,  )  Thus from. (2)  lk-s* » y * ) M i X I K : y * ) l l n  n  s=u  1  oo  <  I B/2 s=y n ' k=l  Let f = Q  Q  + *.  = B  aS  ff  1 - 2  1 — 2 M  < c B/2  a U  J  C3B v. k  ar *  k  By ( U ) , A + ( f )  •  s a t i s f i e s the conditions  o f Theorem 5  o f t h e p r e v i o u s C h a p t e r , and we conclude t h a t * b e l o n g s t o H (K ) P 1 i t 1 where = a r ^ 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 o f t y p e 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 ) . T h e r e f o r e from Lemma 1 , 0 n b e l o n g s t o t h e 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 t h e c l a s s  (?) with K  ±  <_ hk + c^B.  The second d o n e l u s i o n o f t h e theorem f o l l o w s d i r e c t l y from the  inequality  N  I I f : L (R )|| <_A + I B / 2 s=l  d S  = A + cB  P  and t h i s completes t h e p r o o f . We now p r o v e t h e f o l l o w i n g r a t h e r u n u s u a l theorem w h i c h  states  t h a t c e r t a i n m i x e d 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 H o l d e r c o n d i t i o n and a norm i n e q u a l i t y where we m i g h t e x p e c t t h e s e p r o p e r t i e s  o n l y f o r t h e non-mixed d e r i v a t i v e s  by w h i c h we d e f i n e o u r space. -+  THEOREM 1. M^= the  L e t t h e f u n c t i o n f b e l o n g t o t h e c l a s s H ( M ) where  = M  r  R  = M, and l e t t h e n o n - n e g a t i v e i n t e g e r s 6^, ***» $  n  satisfy  inequality n  SkAk < 1 '  I  k=l  Then t h e p a r t i a l d e r i v a t i v e s  Bi + , , * + e n  3 • • l  ••  I  Pk k>'  3  x  f  x  n  ^  e x i s t on R  t o t h e c l a s s H (M ) where P r  i =  r  i  (  "  l  / r  M' = ••• = M' = M' < UlIf : L^R")!! + cM. 1 n — P 11  For t h e s e d e r i v a t i v e s we have t h e i n e q u a l i t y  r  N  N  : L (R )|| < | |f : L (R )||  +  cM.  I n ( 7 ) and (8) t h e c o n s t a n t c does n o t depend f o r M.  and b e l o n g  -»• n 1 S i n c e f e H*"0M) w i t h M, = ••• = M_ = M, t h e n A+(f)p < dM^—R P 1 v  PROOF.  v  k  k  x  -»• f o r any v.  Let v  fe  s/rv = 2  ( s = 0, 1 , * " * ) , and denote b y g^ =  g g g f x ^ , * * * , x ) an e n t i r e f u n c t i o n o f t y p e  i n the v a r i a b l e x^  n  which  satisfies | |f - g  : L ( R ) | | <,dnM/2 . s  s  p  Then t h e f u n c t i o n f i s r e p r e s e n t e d b y t h e s e r i e s c o n v e r g i n g i n L ( R ) n  p  norm s=l  °0  *0  =  %=s - g _  »  s  8  x  where ||Q  g  : Lp(R  n  < 3dnM/2  )M  s  (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 g e t  g ...+e 1+  g ... p  n  3  1+  f  Sx^  '••3x  +  B1+-«»+Bn  n  3  Q 1  n  *"3x  n n  0  »  3  s=l 3x * " 3 x 1  x  where  *i ---*n 1  and  n  p  Q  p  n n  s  U8 3  O  1  3x, *«*3x , n 1 n 1  P  3dnM  s  2  p  T  s[( E =(3dnM)2'  ( s  ?  S /r ) k  k  s  Pk/rk)- ] 1  ( s = 1, 2, • • • ) .  1  Line (5) insures 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 t h e  theorem i s p r o v e d b y a p p l y i n g Lemma 2 t o t h e s e r i e s ( 9 ) . r We now d e f i n e a norm on t h e space H  p >  and a p p l y t h e p r e v i o u s  theorem t o show completeness o f t h e normed space.  R e c a l l that through-  out t h i s Chapter we a r e u s i n g t h e H B l d e r c o n d i t i o n on t h e f i r s t d i f f e r e n c e ( f o r a < l ) o f t h e p - t h p a r t i a l o f f t o d e f i n e t h e space. The norm o f t h e space H ( G ) w i l l be g i v e n b y p  n  l | f : l£(0 )|| = | | f : L ( G ) | | • M n  p  n  (10)  f  where M- i s t h e infimum o f c o n s t a n t s M such t h a t f E H ((T ,M) where r P n» = ••• = M  = M; i . e . M  R  i s t h e i n f o f c o n s t a n t s M such t h a t t h e  f  following i n e q u a l i t y holds f o r a l l i :  ! ! A x y i \ h ) : L (G i i x  if  p  n j T i  )||<Mfh!  a i  < 1, o r  if a  ±  = 1. ->•  THEOREM 2. PROOF.  H  r  p  i s complete.  L e t { f } be a sequence o f f u n c t i o n s i n H m p ||f  k  - f j : Hp|| * 0  as k , J *  f o r which ( l l )  h9 By t h e p r e v i o u s theorem t h e 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 fv  a f : L (R )|| < e | | f k - f , 3x " " " J  S  s  k  J  3x*  n  S  P  : <| |.  '  (12)  k  x  Thus 8 f  8 f i : L (R )|| 3xf  S  S  k  0  n  af  as k„1 + »,  (13)  p  x  and from ( l l ) and (13) we conclude t h a t t h e r e e x i s t s a f u n c t i on f e L ( R ) having p a r t i a l derivatives f ^ p x^ such t h a t  ( s = 0, • * *, p. ; i = 1,* * * ,n)  n  3 f  1  3 f  s  s  m  L_.(R )! |  0  n  3x*  Sx*  as  m  «.  P  ? Since the functions f  m  a l l b e l o n g t o H , each f ^ s a t i s f i e s a f i r s t o r p  second d i f f e r e n c e c o n d i t i o n on t h e p - t h d e r i v a t i v e i n x^: e.g. f o r x  a  ±  < 1,  i'V *»v • V " f  Thus l e t t i n g m  h>  Rn)  ± M M  f  a i  <* - 1 .  2  » •••• » ) •  ^ » we have t h e same c o n d i t i o n s a t i s f i e d f o r t h e -*•  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 quantity  i s s u f f i c i e n t l y small.  x  x  Then l e t t i n g J -*•<*> ve have t h a t  m^  x  ±  i s s u f f i c i e n t l y s m a l l , and we c o n c l u d e t h a t -»•  IIf I I  m  — f :H P'  I | -> 0  as  B  +  ».  1  T h i s completes t h e p r o o f . r We now t u r n t o some s i m p l e imbedding, theorems f o r t h e c l a s s H r r w h i c h demonstrate t h e r e l a t i o n s h i p between H  and W  s  for  i n t e g e r v a l u e s o f t h e parameter r . p  By d e f i n i t i o n , f b e l o n g s 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 o f o r d e r p b e l o n g t o L ( P r ) . (We o f course r e f e r t o t h e d e f i n i t i o n 1 x  of f ^  as g i v e n i n Chapter 2.  See Chapter 8 f o r a d i s c u s s i o n o f t h e  equivalence t o the generalized d e r i v a t i v e . )  II*  :W  P  P» l x  I!  ||f  = | | f : L (H )|| • P n  The norm i s g i v e n by  (p)  1  : L (R )||. n  p  Note t h a t t h i s one d i m e n s i o n a l d e f i n i t i o n c o r r e s p o n d s t o t h e g e n e r a l case o f t h e S o b o l e v spaces as d e f i n e d i n [ l U ] , b u t t h a t t h e n-dim-> p r. e n s i o n a l W ( d e f i n e d a n a l o g o u s l y t o H ) i n v o l v e s assumptions on o n l y p  the  p  non-mixed p a r t i a l s f  THEOREM 3.  (Pi) x  i  The f o l l o w i n g imbeddings  r r' H •*• H P» ! P,x  ,  x  x  hold:  0 < r < r ;  Note t h a t "by an imbedding v e mean n o t o n l y a s e t t h e o r e t i c a l i n c l u s i o n , b u t a l s o a norm i n e q u a l i t y .  Thus  M + M*  h o l d s i f and  o n l y i f M c M , and t h e r e e x i s t s a c o n s t a n t 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 ( 1 7 ) a r e t o be e x p e c t e d o f any r e a s o n a b l e g e n e r a l i z a t i o n o f Sobolev spaces.  The r e l a t i o n ( l 6 ) t h e n shows t h e completeness  t h e d e f i n i t i o n o f H* of t h i s theory.  ( 1 5 ) stems from a l l o w i n g a = 1 i n  Thus w h i l e H  P p  i s the wider class f o r integer values  o P+ i s l a r g e r t h a n H„ f o r anv a > 0. p P a  o f p, W  following transitive  That i s , we have t h e  relation:  t h e end r e s u l t o f w h i c h i s a l r e a d y g u a r e n t e e d b y The p r o o f s o f p r o p e r t i e s ( l U ) - ( 1 7 ) a r e q u i t e easy and depend on Theorem k o f Chapter 5 f o r t h e s e t i n c l u s i o n s and Lemma 1 o f Chapter 3 and Lemma 1 o f Chapter 5 f o r t h e norm i n e q u a l i t i e s .  As an example o f  t h e p r o o f s , we show ( 1 5 ) and ( 1 6 ) . PROOF OF ( 1 5 ) . Chapter 5 ,  Thus f e  Let f e W . p,x^ p  and f u r t h e r m o r e l P.*l  Then f , t^t) e L ( F ) and b v Lemma 1 o f j. n  p  52 and we c o n c l u d e ( 1 5 ) . PROOF OF (16).  Let f e H  where r = p + a.  f  v. l  e  x  L_(R ) and  w  e  p  Since 0  conclude t h a t f e W  p  P,x  . Now b v Lemma 1 o f x  C h a p t e r 3,  ! L (Rn)  p "i  c  lH  f 8  V^^! +  C  M  2 f  whence  Mf  :W  S',x  II id+^JMf  < c||f : H j  t X i  ||.  T h i s completes t h e p r o o f o f ( l 6 ) .  : L (R )|| + c M n  p  < p,  ~~  ,  x  2  f  CHAPTER SEVEN AN IMBEDDING THEOREM FOR THE SPACE H  p  W h i l e Theorem 3 o f t h e p r e v i o u s Chapter showed r e l a t i o n s h i p s P P _ ,H , and W , i t c o n s i d e r e d o n l y spaces P.*!' P , x ' p,x » d e f i n e d o v e r a l l o f R h a v i n g t h e same v a l u e o f p. The f o l l o w i n g among t h e c l a s s e s H  r  v  x  x  n  theorem, t h e r e a l c u l m i n a t i o n o f t h i s work, g i v e s c o n d i t i o n s on t h e -+-  parameters r , ***, r x  n  f o r t h e imbedding o f H  p  i n t o s i m i l a r spaces  d e f i n e d o v e r l o w e r d i m e n s i o n a l subspaces o f R* and h a v i n g d i f f e r e n t values o f p.  I f we p u t 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 t h e S o b o l e v imbedding theorem f o r W . p  (See [ 2 ]  and [ 1 5 ] . ) THEOREM 1.  Suppose 1 <_ p <_ p  > 0 , i * ,1,  < «° and  n. F o r  1 < m ««n l e t k m  = 1 - l / p ( I l / r ) + l / p ' ( I l / r ) > 0. i=l i=l ±  ±  •f +  n  I f f ( x , * " , x ) b e l o n g s t o t h e c l a s s H (M) on R , then f o r f i x e d 1  x x  m + x  ,**",x  n  n  p  t h e f u n c t i o n f c o n s i d e r e d as a f u n c t i o n o f t h e v a r i a b l e s  l****» Hl b e l o n g s t o t h e c l a s s H ,(M ) on i f where x  p  ,5=1  J  w i t h t h e c o n s t a n t y depending o n l y on n , and 6 depending o n l y on  n , m, r ^ , •**, r  n  Thus we have t h e imbedding  #  Hp(B. ) * H ^ R * ) . n  PROOF.  1  L e t f e Hp(M) and l e t v ^  * 2  1  f o r s = 0, 1, '' *.  S  Then b y  Theorem 2 o f Chapter 5,  (*I V  ^Wpi ^ Since  / 2  k=l  '•  - v ^ ( s ) i s a f u n c t i o n o f s , denote b y g  of t y p e  i n the variable x  satisfying  k  : Lp(R )|[ < (d I M )/2 k=l  ||f - g  an e n t i r e f u n c t i o n  g  n  S  k  .  Then t h e f u n c t i o n f can be r e p r e s e n t e d b y t h e s e r i e s (which converges i n t h e L ( R ) norm): n  00  OO  f = g  0  +  I  s=l  ( g - g _!) = Q s  8  +  0  I %  .  s=l  Then ||Q  : L _ ( R ) | | <^ ( 2 d J M . ) ^ " k=l n  8  (s = 1, 2,  1  S / rir  and Q  s  i s ah e n t i r e f u n c t i o n o f t y p e 2  i n t h e v a r i a b l e x^.  C o n s e q u e n t l y , b y l i n e s (10) and (12) o f Chapter k f o r t h e subspace R  m  of R  o f p o i n t s u , * ** .u^ . x ^ , • • • , x  n  x  norm o f Q ||Q  8  s  i nR  m  n  for fixed  x ^ , ' " . ^ the  satisfies  : Lp.tR*)!! =  = (J  •••/  |Q (u ."-,u ,x g  1  m  n + 1  ,---,x )| n  P  du —du 1  f f l  )  1 / p  '  56  <_2  (n  n  v  K  )  1  /  P  ,  | | Q  :  S  L  m+1  1  .<F»)|| *  m+1  p  <22n+2(dK)/2 \ 8  k=l  k  Therefore  s=l < 2  2 n + 2  s=y d K k-1  J - i — s=u 2 ^  k  S  <  ~  (c l M ) / 2 k-1  P K m  k  k  S i n c e QQ +  v h e r e t h e c o n s t a n t c depends on r and < . m  £ Q- i s o f t y p e 1 8  u AM no g r e a t e r t h a n 2 i n the variable x , x  s=l  k=l  p  1  By Theorem 3 o f Chapter 5, t h i s means t h a t f - g as a f u n c t i o n o f 0  Xi »** * ,x. b e l o n g s t o t h e c l a s s H ^ " ^ ( R , M j ) where 8  1  m  n  with 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 Now  the function  g  Q  e ^ ' ( M ) f o r any f i x e d  x  5  R n )  n  i s o f t y p e u n i t y i n a l l t h e 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 V  , **,x . #  m + 1  H 1H  f  5  V  R n )  ll  +  d  Ik • M  and by l i n e s (10) and (12) o f C h a p t e r h we have t h a t ||g  : L ,(P»)H < 2 ! | g  : L ,(K )||  n  0  p  <2  2 n  < 2  2 n  ||g  0  :  n  0  p  1^)11  ( | | f : L (R* ) 11  + d f ^ ) .  1  k=l  p  Thus b y Theorem h o f 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 t h e sum o f t h e c o n s t a n t s  and  cannot exceed  n Y  ||f  : Lp(* )(| + B I M n  k  ,  k=l w h i c h shows t h a t f e H »(M ) f o r any f i x e d p  M* < | | f : Y  l^mW  + B[M  x m + x  »***»  x n  where  ,  k  and t h i s completes t h e p r o o f .  We remark tkat this transformation f*om r t o s^m) v i a tc i s x  transitive.  I n d e e d , i f we have numbers r , • " , r x  00  and t h e n chose numbers p  spaces R" and R  J  w i t h j < m < n , then  n  and p  > 0  and p w i t h  ii  i  and 1 <_ p <_  m  w i t h p <_ p  t  tt  <_ p  and  t h e t r a n s f o r m a t i o n from (p,R )  t o (p'jR ) and t h e n from ( p ' j R ) t o (p",R^) r e s u l t s i n parameters 111  01  (<1)» **** s Cj) w h i c h  S X  n  t o (p",R- ). 1  can be o b t a i n e d b y g o i n g d i r e c t l y from (,p,R ) n  58 Nov i t i s p o s s i b l e t o demonstrate t h e f o l l o w i n g r e s u l t w h i c h shows t h a t t h e c o n d i t i o n ( l ) may n o t be weakened. I f 1 - l / p ( ^ 1 / ^ ) > 0, 1 < p , and e > 0, t h e n f o r any p' and m w i t h 1 < p _< p' and 1 <^ m f _ n , t h e r e e x i s t s a f u n c t i o n f belonging s =  t o H** b u t n o t ( f o r any c o n s t a n t M) t o H_»(M) where  (s (m),* *,s _ (m),s (m)+e,s (m) "«,s (m)). ,  1  i  1  i  i+1  f  in  I n E'9] pp. 2T0-27H ( o r pp. 30-33 i n t h e t r a n s l a t i o n ) N i k o l ' s k i i  gives  an example o f a f u n c t i o n s a t i s f y i n g t h e above statement f o r t h e case i  m = 1 and p  = ».  B u t b y t h e 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,R H.to (»,R) s u f f i e i e s t o c o v e r a l l t h e cases (p* .R™) w i t h n  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 t h e H-spaces. the c o n s t r a i n t t h a t <  m  T h i s i s r a t h e r unexpected because  > 0 would appear t o be d e r i v e d s o l e l y from t h e  t e c h n i c a l i t i e s i n t h e p r o o f f o r convergence o f t h e s e r i e s (.5) o f entire functions.  This apparently  demonstrates t h e p r e c i s i o n o f t h e  approximation by e n t i r e f u n c t i o n s . At t h e o u t s e t i t was mentioned t h a t t h e s e r e s u l t s a r e d i f f e r e n t i n form from t h o s e o f S o b o l e v f o r i n t e g r a l parameters r . B u t t h e y a r e 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 S o b o l e v ' s theorems i n t h e sense t h a t the non«*mixed p a r t i a l s d e f i n e d h e r e a r e a c t u a l l y t h e g e n e r a l i z e d d e r ivatives. Suppose t h e f u n c t i o n f ( x , * , x _ ) has t h e non-mixed p a r t i a l , ,  1  o r o r d e r 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  (  1)  P X  l  absolutely  f^  c o n t i n u o u s 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 x  2»  *n)  w  h  i  c  "belongs t o L ( R ) f o r almost a l l x , * " » x .  h  If  n  p  2  n  <P i s any f u n c t i o n w h i c h , t o g e t h e r 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 t c o r d e r p , i s c o n t i n u o u s snd v a n i s h i n g on t h e boundary o f t h e cube Q c c o n s i s t i n g o f p o i n t s ( x , * " , x ) which s a t i s f y 1  n  — ± f. i» x  b  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 b y p a r t s we get t h e e q u a t i o n  b  /  l  9 p f  a-^  o l  J b  (-D  p~ d x i -  f  dx^  3  °  f  —T  *  B.^  3x^  w h i c h h o l d s f o r almost a l l x » * " » XJJ. 2  After integrating with r e -  s p e c t t o ( x g * * * * , ^ ) we have  »* P  j  (f fl  p+1  -+ 3x  (-1)  «P*)dx = 0,  p x  3 f p  and we conclude t h a t * =  i s indeed the generalized d e r i v a t i v e . 3  x  l  C o n v e r s e l y , l e t $ e Lp(.R ) be t h e g e n e r a l i z e d d e r i v a t i v e o f f i n t h e v a r i a b l e x-^.  As shown b y S o b o l e v  ([lU], s e c t i o n 5) t h e r e  c o r r e s p o n d s t o t h e f u n c t i o n f an a v e r a g i n g f u n c t i o n  ("mollifier")  f u e L ( R ) depending on h w h i c h has c o n t i n u o u s p a r t i a l d e r i v a t i v e s n  o f any o r d e r such t h a t | |f - f  " and  \  3  p 3  x  l  *  *  n  : L p ( R ) | | •* 0 n  h  as h •+ «  n e Lp(,R ) where  3 f *l! 3x^ p  : L p ( R ) | | + 0 as h + n  But b y Lemma 2 o f Chapter j> we t h e n c o n c l u d e t h a t f may be changed on a s e t o f measure z e r o s o t h a t 3 f p  0  3X-,  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  = r and f o r spaces d e f i n e d o v e r R , o f some theorems n  n  of Sobolev.  I n f a c t , l e t t h e f u n c t i o n f e L ( R ) have a l l g e n e r a l i z e d n  d e r i v a t i v e s o f o r d e r r ( r an i n t e g e r i n t h i s i n s t a n c e ) w h i c h b e l o n g to L ( R ) . n  p  Then t h e non-*nixed p a r t i a l d e r i v a t i v e s are t h e u s u a l  der-  i v a t i v e s d e f i n e d h e r e , and b y Lemma 1 o f Chapter 5»  ||A  X x  l  (f  ( x x  "  X )  n  l  ih||A  , h ) : L (R )|| p  (f ^.h) (  X i  <2h||f  (  > x  : LpCP")!!  : L (R )|| n  p  1  and thus f b e l o n g s 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  function f , considered  Consequently, the  on t h e l i o e a r subspace x ^ ,  x ^ belongs  t o H «'(M*) where s • ( s , * * * , s ) w i t h p  s = ric  m  = r - m/p  f  + n/p > 0,  and <_ Y ! \? : L p ( R ) | | + SnMj^ n  , ( l < _ i <ti).  Thus i f we l e t s = p + a , p an i n t e g e r and 0 < a < 1, then t h e 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 o r d e r p w i t h r e s p e c t t o each x j w h i c h belong t o L p i f R ) . 8 1  We may o b t a i n e s t i m a t e s  f o r the Lp^R  111  ) norms  o f t h e s e d e r i v a t i v e s b y e s t i m a t i n g t h e s e r i e s (h) u s i n g (5) t e r m w i s e .  6l Furthermore  i fr  = *** = r  ±  R  = r i n Theorem 1, we have o n l y t h e  single condition that n K  m  =  m  1  +  pr  —  p r  >  0  or e q u i v a l e n t l y ,  P  i  <  mp n-rp  •  I t now f o l l o w s i m m e d i a t e l y t h a t t h e 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  F o r a s i m i l a r r e s u l t f o r t h e mixed p a r t i a l d e r i v a t i v e s we Theorem 1 o f Chapter  6.  recall  CHAPTER EIGHT THE CLASSES Hp*(G ), Wp*(G ), Bp*(G ) n  n  n  I n t h i s f i n a l Chapter v e make b r i e f comment on t h e e n t i r e spectrum o f f r a c t i o n a l o r d e r S o b o l e v s p a c e s , 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 a l r e a d y mentioned and t h e p r o p e r t i e s t h e r e o f . We a g a i n assume t h a t 1 < p < « and t h a t G i s an open s u b s e t o f — — n  R. n  the  Then f o r n o n - n e g a t i v e i n t e g e r s p , f = f ( x »* * * ,x ), b e l o n g s t o x  n  c l a s s Wp(G ) i f t h e f o l l o w i n g norm i s f i n i t e : n  II*  « W (G )|| p  n  [||f : V n M I G  P  • El|ftp> t V G n ) M P l 1 / P  where t h e sum i s t a k e n o v e r 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 o r d e r p. Now f o r any r > 0 we examine t h e f r a c t i o n a l o r d e r W, B, and H spaces w h i c h p r o v i d e two b a s i c c l a s s i f i c a t i o n s f o r t h e c o n t i n u o u s The f i r s t i s t h e f a m i l y o f c l a s s e s W5(G ) w h i c h i s n a t u r a l l y  functions.  completed b y t h e c l a s s e s B ( G ) . p  n  The second c l a s s i f i c a t i o n i s made up  of t h e c l a s s e s H p ( G ) as a l r e a d y d e f i n e d , where we make t h e c o n v e n t i o n n  -V  that i f r  ±  = •** = r  n  = r , t h e n we w i l l w r i t e Hp"(G ) f o r H*(.G ). n  n  L e t t i n g r = p + a as b e f o r e , i f o 4 1, t h e n f b e l o n g s t o Wp(G ) n  i f t h e f o l l o w i n g norm i s f i n i t e :  l|f * WJCGJM = f | |f : y o n > | | P * (P), x Jp), .,P jf ( x ) - f (y)| U  +  ^  i ° h G  n  —  ;  ^  l - yl  ** &  1  x  /  x  where t h e sum i s t a k e n o v e r 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 y P  I f a * 1, t h e n f b e l o n g s t o p ( G ) i f t h e norm ( l ) i s f i n i t e . w  n  To d e f i n e B p ( G ) i f a 4 1, t h e n Bp(C» ) = Q  t  n  WJJCOjj).  and i f a = 1,  f b e l o n g s t o B ( G ) i f t h e f o l l o w i n g norm i s f i n i t e : p  ||f  n  : Bj(G )|| = [ ||f : n  |f + IJ G  where G  ( p )  dxj n  G  Lp( n>M P G  (x) - 2f •  ( P )  +  ((x y)/2) — +  ( p )  n  (y)|  P l  /  p  dy]  I* - yl  n,x  i s the setof a l l points y e G  n x  • f  such t h a t ( x + y ) / 2 c G , n  and where t h e sum i s a g a i n t a k e n o v e r 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 o f o r d e r p. The c l a s s H ( G ) i s d e f i n e d as i n Chapter 2 w i t h norm as g i v e n i n p n r  C h a p t e r 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 s u b m a n i f o l d  of G . n  We now s t a t e t h e w e l l known  g e n e r a l 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  Wp( Gn) - Wpt ( Gm ) P  P  n  where p  i  m  i  i s the integral part 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 c o n s i d e r t h e c l a s s H p ( G ) , we have a l r e a d y p r o v e d Theorem 1 n  o f Chapter 7 f o r R  n  which s t a t e s f o r G  that  r ' = r - n/p + m/p*  > 0  , 1  < p'  to  «  Now r e c a l l t h e comment a t t h e end o f C h a p t e r 6 w h i c h r e f e r s t o t h e 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  w h i c h h o l d s f o r p an i n t e g e r . And a t t h e end o f t h e p r o o f o f Theorem 1 o f Chapter 7 we s t a t e d t h e t r a n s i t i v e n a t u r e o f t h e imbedding ( 3 ) .  Two  c o n s e c t u t i v e t r a n s i t i o n s from (p,n) t o (p',m) and t h e n from (p',m) t o (p , j ) may be r e p l a c e d b y one d i r e c t t r a n s i t i o n from (p,n) t o (p I n t h e s e s e n s e s , t h e n , t h e f a m i l y o f c l a s s e s H^(G ) i s c l o s e d w i t h P n r e s p e c t t o imbeddings. P We ask i f t h e c l a s s e s p ( W  G R  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  p o s s e s such a c l o s e d p r o p e r t y w i t h r e s p e c t t o imbedding theorems. f o r t u n a t e l y t h e y do n o t , as i n d i c a t e d b y t h e f o l l o w i n g THEOREM 2 .  Un-  theorem.  I f 1 < p < p' <_» and r ' = r - n/p + m/p', t h e n t h e imbedding  h o l d s o n l y under t h e 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 , t h e n we have t h e f o l l o w i n g imbeddings w i t h o u t exception, t  • W£(G ) n  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 , p ( ^ ) = ^ p ^ ^ r r S  G  n  a l s o have t h a t f o r a l l r , B ( 0 ) = 2 ^ n ^ * W  2  G  7 1 x 1 1 3 t  h  #  W  e  n  e  n  ^Port  ° the  8 1 1 0 6  f  B c l a s s e s , w h i c h d i f f e r from t h e W c l a s s e s o n l y f o r i n t e g r a l r , i s shown b y t h e f o l l o w i n g two theorems, t h e f i r s t o f w h i c h shows t h a t t h e B c l a s s e s p o s s e s a c l o s e d system o f imbeddings as do t h e H c l a s s e s . THEOREM 3.  I f 1 <, p <_ p' <,  00  *p(c ) n  and r ' = r - n/p + m/p' > 0, t h e n  *BJ:(OB).  F i n a l l y we f i n d t h a t t h e completeness o f t h e W c l a s s e s w i t h  respect  t o imbeddings can be a c h i e v e d o n l y b y t h e 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 f1 < p < p WJ(0 F  n  <_«> and r  = r - n/p + m/p  > 0, then  ) - B*\(G). p m  Jbr f u r t h e r i n f o r m a t i o n  on t h e H, B, W c l a s s e s f o r any r > 0  we c i t e t h e f o l l o w i n g r e f e r e n c e s .  F o r t h e use o f 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 t h e W c l a s s e s and an i n t e r e s t i n g g e o m e t r i c a l t r e a t m e n t o f imbedding t h e o r e m s , see S o b o l e v and N i k o l ' s k i i , [ l 6 ] , s e c t i o n 1.  F o r an a n a l y t i c d e f i n i t i o n o f t h e 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 t h e d e f i n i t i o n o f 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 g i v e s a s u r v e y o f r e s u l t s on H and P »i P i W , w h i l e s e c t i o n s 3 and h g i v e a thorough resume o f p r o p e r t i e s o f P ,XJ 7 r t H ( G ) , B ( G ) and W ( G ) . M n a l l y , S l o b o d e t s k i i has p u b l i s h e d many x  p  n  p  n  p  n  works on t h e W c l a s s e s as d e f 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 [ 1 3 ] .  66 BIBLIOGRAPHY  1.  A h k i e z e r , N. I . L e c t u r e s i n t h e Theory o f A p p r o x i m a t i o n s , 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 S o b o l e v S p a c e s , Seminar N o t e s , U n i -  v e r s i t y o f B r i t i s h C o l u m b i a , Vancouver.  1968.  3.  H a r d y , 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, D o k l . Akad. Nauk SSSR, (1959h 1187-1190. ( R u s s i a n ) .  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 Func t i o n s , I z v . Akad. Nauk SSSR S e r . Mat. 2k 865-882. (Russian).  6.  N i k o l ' s k i i , S. M. E x t e n s i o n o f F u n c t i o n s o f S e v e 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) E n g l i s h T r a n s l . AMS T r a n s l . (2) 159-188.  126  (i960),  (1956);  83 (1969),  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 Several V a r i a b l e s , D o k l . Akad. Nauk SSSR, (9) il9kQ) 1533-1536. ( R u s s i a n ) .  59  8.  t  N i k o l ' s k i i , S« M. Imbedding Theorems f o r F u n c t i o n s v i t h P a r t i a l D e r i v a t i v e s C o n s i d e r e d i n V a r i o u s M e t r i c s . I z v . Akad. Nauk SSSR S e r . Mat. 22 321-336; E n g l i s h T r a n s l . AMS T r a n s l . (2) 88 ( 1 9 7 0 ) , 1-U0.  (1958),  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 E x p 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 t h e Theory o f D i f f e r e n t i a b l e F u n c t i o n s o f S e 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 ( 1 9 5 1 ) , 2 ^ - 2 7 8 ; E n g l i s h T r a n s l . AMS T r a n s l . ( 2 ) 80 1-38.  (l96l),  10.  N i k o l ' s k i i , S. M.  Of One P r o p e r t y  o f t h e C l a s s H_*, Ann. t h i v . S c i .  B u d a p e s t , E o t v o s S e c t . Mat. 3-k 11.  (Russian).  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 A p p r o x i m a t i o n Theories f o r D i f f e r e n t i a b l e Fiinctions o f Several V a r i a b l e s , R u s s i a n Math S u r v e y s , 1961,  l6 (5)  12.  (1960/1961), 205-216. 55-10U.  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 C l a s s e s o f F u n c t i o n s o f S e 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. 33 261-326; E n g l i s h T r a n s l . AMS T r a n s l . ( 2 ) 80  (75) (1953), (1969), 39-118.  67 13.  S l o b o d e t s k i i , L. N. On S o b o l e v ' 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 R e g i o n a l 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 E q u a t i o n s , D o k l . Akad. Nauk SSSR, (Russian).  118 (2) (1958), 2U3-2U6.  lh.  S o b o l e v , 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 M a t h e m a t i c a l Physics. American M a t h e m a t i c a l S o c i e t y , P r o v i d e n c e , R . I . 1963.  15.  S o b o l e v , S. ,L. k  16.  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.  (U6) (1938), U71-VT9.  (Russian).  S o b o l e v , S. L. , and S. M. N i k o l ' s k i i , Imbedding Theorems, P r o c . F o u r t h A i l - U n i o n Math. Congress ( L e n i n g r a d , 196l), I z d a t . Akad. Nauk SSSR, L e n i n g r a d , 227-2*42; E n g l i s h T r a n s l . AMS Transl.  I963. (2) 87 (1970), 1U7-173.  17.  Y o s i d a , Kosaku.  18.  Zygmund, A.  Functional Analysis.  Springer-Verlag,  Trigonometric S e r i e s , V o l . I I .  P r e s s , London.  1959.  Berlin.  I96U.  Cambridge U n i v e r s i t y  

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