lorentz spaces youngs inequality convolutions thesis

7/27/2019 Lorentz Spaces Youngs Inequality Convolutions Thesis

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AN EXTENSION OF YOUNG'S INEQUALITYby

CHARLES DENNIS SISSEL, B.A.A T H E S I S

INMATHEMATICS

Submitted to the Graduate Facultyof Texas Tech University inPartial Fulfillment ofthe Requirements forthe Degree ofMASTER OF SCIENCE

Approved

Accepted

A u g u s t , 1 9 8 8


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/ f r-/ ^ A) o ^

:?l 1 .sr^

A C K N O W L E D G E M E N T S

I would l ike to thank Dr . Char les Kel logg fo r h i s adv ice in se iec t ing the top icfo r t h i s t h e s i s , I wo u l d a ls o l ik e t o t h an k Dr . Ha ro l d B en n e t t an d Dr . Da v i dW e i n b e r g f o r t h e i r s u p p o r t a n d i n s t r u c t i o n d u r i n g m y g r a d u a t e c a r e e r a t T e x a sT e c h U n i v e r s i t y .

I wo u l d li ke t o ex p re s s m y d eep ap p rec i a t i o n t o Dr . B e r n a rd M a i r for h i sp a t i e n t d i r e c t i o n a n d c r i t i c i s m t h r o u g h o u t t h e p r e p a r a t i o n o f t h i s t h e s i s . I h o l dh i m i n t h e h i g h e s t r e g a r d a s a p r o f e s s i o n a l m a t h e m a t i c i a n a n d a s a p e r s o n a lfriend.

I o we m y d eep es t t h an k s t o m y f a t h e r an d m o t h e r fo r t h e i r s u p p o r t an da f i rm a t i o n . I t was t h e i r en co u rag em en t wh i ch m ad e i t p o s s i b l e fo r m e t o ach i ev et h i s wo rk . I w i s h t o d ed i ca t e t h i s t h e s i s w i t h g ra t i t u d e i n h o n o r o f J e s u s C h r i s t ,t h e L o r d .

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C O N T E N T SA C K N O WL E D G E ME N T S i iABSTRACT ivLIST OF FIGU RES v

I . INTROD UCTION 1II. THE DISTRIBUTION FUNCTION AND NONINCREASINGR E A R R A N G E M E N T S 7

III . AN EXTENSION OF YOUNG'S INEQUALITY FORCONVOLUTIONS OF FUNCTIONS IN THE LORENTZSPAC ES ." 29R E F E R E N C E S 68

111

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A B S T R A C T

T h e c l a s s i ca l Yo u n g ' s In eq u a l i t y fo r co n v o l u t i o n o n t h e L^ s p a c e s s t a t e s t h a t\/ f e Lp an d g G Lq,

1 1 / * 5 ' l l r < l l / l l p l l ^ l l q ,w he re 1 1 . The a im of th i s thes i si s to p resen t a d i rec t p roof o f a genera l iza t ion o f the c lass ica l Young ' s Inequal i tyfor fu n c t i o n s i n L o r en t z s p ace s . In p a r t i c u l a r , we p ro v e t h a t co n v o l u t i o n s a t is f ie st h e restricted weak condition,

| | / * ^ | | p . o o < i ^ | | / l k l l l ^ l k l ,fo r - s o m e co n s t an t K , i n d ep e n d en t of / a n d g, wh en ev e r 1 / p = 1/u -\- l/r 1.H a v i n g d o n e t h i s , w e a p p l y a n E x t e n s i o n o f t h e M a r c i n k i e w i c z I n t e r p o l a t i o nT h eo re m for s u b a d d i t i v e o p e ra t o r s o n Xp_g w i t h t h e s e en d p o i n t s , ( s ee [7 , p ag e1 9 7 ] ) , t h e r e b y o b t a i n i n g t h e r e l a t i o n

| |/ * 5 r | |p , g < i ^ ll /l lu . g' li ^ ll r ,, " , V g ' , / , 1 < q',q" < o c ,wh e re K i s a co n s t an t i n d ep en d en t o f / an d g.

IV


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L I S T O F F I G U R E S

2.1 f = X ^ 122.2 f = X-'" 173.1 f = ^ A * '^B 343.2 = ^i^uh * ^Jiuj^ 473.3 A Bo und For /"-() 48

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C H A P T E R II N T R O D U C T I O N

In the discussion which foUows, we define {M,^i) an d [N,v) to be 1 , T i s a subl inear opera tor mappingS{M) in to M{N). For g a fixed, measurable funct ion, denote Tg{f) = f * g.

Let f ^ Lp an d g G Lq, 1 < p, 9 < 00, wh ere by assu m ptio n, 1/p + 1/q >1. The classical Young's Inequal i ty for convolut ion states that Vr,p, and q sucht h a t 1 / r = l / p + 1/q -1,

l l / * 5 r | | r < | | / | | p | | ^ l k ( 1 . 1 )The object of this paper is to prove an extension of Young's Inequal i ty for con-volut ion as an operator on the larger col lect ion of spaces Lp^q. Assume tha t u isf ixed, 1 < l < 00. Motivated by the hypotheses given for the classical Young'sInequal i ty , we sha l l assume throughout the thes i s tha t for i=0 and 1, Pi,u, an dri sat isfy the relat ion 1/pi = 1/u -\- l/r^ 1. As usu al, (cf. [7]), we m ake th efollowing definitions.

For all a > 0, deine the setEa. = {x'. \f{x)\ > a}.

Deine the distribution function o ,f.^^{a) = PL{E^), a > 0 .

For t > 0, define the nonincreasing rearrangement oi / , /^ asf;{t) = inf{a:f.,^{a)

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For f ixed p a n d q, 1 < p < oo, 1 -< q < oc, define

i i / iu={r [ ' / - ( .T h e fu nc t io na l I j. llp, , i s order preserving in the sense that i f / and g a r e m e a -su rab le fu r ic t ions sa t i s fy ing

\f(x)\ < \g{x)\a.e . , th e n | | / | |p ,5 < | |5 '||p ,q- F u rt h e rm o re , i t m ay be pro ve n tha t for 1 is equivalent to the funct ional

WfW;,, = suptyot'^^m{t).U n d e r t h i s n o r m , Lpq b eco m es a co m p l e t e s p ace fo r ce r t a i n p a n d q. I t may a l sob e s h o wn t h a t t h e s i m p l e fu n c t i o n s a r e d en s e i n Z p , (we d i s cu s s t h e s e n o t i o n s i ng rea t e r d e t a i l i n C h ap t e r I I ) . I t i s s i g n i f i can t t h a t t h e Lp s p aces a r e co n t a i n eda m o n g t h e m .

As u s u a l , a restricted weak type {r,p) o p e r a t o r o n Lpq i s an y s u b l i n ea r o p -e ra t o r wh o s e d o m a i n D co n t a i n s a l l t h e t ru n ca t i o n s o f m em b ers a s we l l a s a l ls i m p l e fu n c t i o n s an d s a t i s f i e s t h e i n eq u a l i t y

| | T / | U < i ^ | | / l i . , i , ( 1 . 2 )


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w h e n e v e r / G S{M) D L, . , i . In their work in [8] and [7; , Stein and Weiss showtha t an opera to r o f res t r i c ted weak type on Lp , , a l so sa t i s f ies Inequal i ty 1 .2 fo ran y m ea su ra b l e func t io n / G I ' r .i - W e wi l l m ak e use o f th i s fac t in th e p roofsp r e s e n t e d i n C h a p t e r I I I .

The p roof o f the foUowing p ropos i t ion i s found in \7 , page 195] .L e t p a n d q b e r e a l c o n s t a n t s s u c h t h a t Lp^q i s a co m p l e t e s p ace , w i t h T a

s u b l i n ea r o p e ra t o r d e fi n ed o n L p , an d E an y m ea s u r ab l e se t of f i ni te m ea s u r e .Su p p o s e t h a t t h e re ex i s t s a co n s t an t C , i n d ep en d en t o f E , s a t i s fy i n g t h e p ro p e r t y

\\TPCE\\p,oo


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for al l / in the domain of T and to L^^^q, w h e n e v e r pt and r^ sa t i s fy the re la t ion]_ _ t 1-tPt Po Pi

a n dn r-o ri

for all t a n d q, 0

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\/v,s, 1 < v,s < o o , an d \/p,u, and r sat is fying 1/p = 1/u l / r - 1 . wh e reK i s a co n s t an t independent of / and g. We ex tend Inequal i ty 1 .6 fo r / and gch a rac t e r i s t i c fu n c t i o n s of fin ite u n i o n s of b o u n d e d in t e rv a l s , b y d em o n s t r a t i n gt h a t / " (< ) an d / (a ) a r e b o u n d e d b y p i ecewi s e l i n ea r fu n c t i o n s . Ap p l y i n g t h edef in i t ion o f the norm | | . | | p ,g , we aga in cons ider the ra t io "^{x), w h e r e x = x{t) 'is a.s u i t ab l e ex p re s s i o n s a t i s fy i n g | a ; | < 1 . E x am i n i n g s ev e ra l c a s e s fo r t h e ex p o n en t sp an d q , we co n c l u d e t h a t :

( I ) For J < 1 , ^ i s bounded on [0,1].( I I ) Fo r ^ > 1 , ^ i s b o u n d ed b y a co n v e rg en t "p " - s e r i e s .

( I I I ) Fo r p = q = oo, "^ 'is b o u n d e d b y 1 .( I V ) F o r ^ = o o , p < o o , ^ i s b o u n d e d o n [0,1] a s a c o n s e q u e n c e of C a s e I .

F ina l ly , we p rove InequaHty 1 .6 fo r A and B arbitrary m eas u rab l e s e t s o ff in ite m ea su re , as fo l lows . A ssu m e th a t \A\ < \B\. Uti l i z ing some bas ic resu l t sf ro m m eas u re t h eo ry wi t h t h e t r i an g l e i n eq u a l i t y fo r Lp^q{M,\\.\\p^q), we provet h a t

| ^ ^ * ^ B | | p , o c < ^ | | ' ^ A | | u , l | | ^ B | | r , l ,for som e con s ta n t K, ind ep en de n t of A an d B . For f ixed , 1 < u < oo , we wi lls h o w t h a t t h e d e t a i l s o f t h e p rev i o u s d i s cu s s i o n i m p l y t h a t Tg i s an opera to r o fr e s t r i c t e d w e a k t y p e s {ri,pi), ^r^ a n d pi sa t i s fy ing the re la t ion 1/pi = l/r^ -\-1/u 1, w i t h c o n s t a n t s d. A p p l y i n g t h e M a r c i n k i e w i c z I n t e r p o l a t i o n T h e o r e mo n Lp^q, i t wiU fol low that

| ; ( / ) IIP,

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Al t h o u g h t h e wo rk co n t a i n ed i n t h e b o d y o f t h i s t h e s i s i s o r i g i n a l , we wi s h t oack n o wl ed g e t h e r ecen t wo rk o f R . A . Hu n t i n p ro v i n g t h e fo l l o wi n g g en e ra l i za t i o no f Yo u n g ' s In eq u a l i t y . In co n t r a s t t o t h e d ev e l o p m en t p re s en t ed h e re i n , Hu n t u s e st h e t o p o l o g i ca l s t ru c t u re o f Xp , , i n h i s d ev e l o p m en t , t h e p ro o f o f t h i s r e s u l t m aybe fo un d in [3 , pa ge s 249 - 274 ] .

T h e o r e m 1 . 1Let0 < 1/p = 1/po + 1/pi - 1, and (1 .7 )

1 < po,Pi < oo , 0


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C H A P T E R I IT H E D I S T R I B U T I O N F U N C T I O N A N D

N O N I N C R E A S I N G R E A R R A N G E M E N T S

2 . 1 B as i c De f i n i t i o n s an d P ro p e r t i e sW e o p en C h ap t e r I I w i t h a d i s cu s s i o n o f t h e d i s t r i b u t i o n fu n c t i o n an d n o n i n -c r e a s i n g r e a r r a n g e m e n t o f a m e a s u r a b l e f u n c t i o n . T h r o u g h o u t , w e a s s u m e t h a t

(M, n , /x ) i s a measure space wi th /x a c r - f in i te measure and M C R . We wi l l u sethe no ta t ion | i4 | when re fer r ing spec i f icaUy to the Lebesgue measure o f a se t A.I t w iU b e u n d e r s t o o d t h a t / an d g a re r ea l -v a l u ed , m eas u rab l e fu n c t i o n s w i t hre s p ec t t o t h e g i v en m eas u re s p ace .

Def i n i t i o n 2 . 1 (D i s t r i b u t i o n Fu n c t i o n o f / )Le t ( M , n ,/Li) be a g iven me as ur e spa ce , wi t h / : M R a m ea su rab le

func t ion . Va > 0 , def ine(i) E^j = {x '. \f{x)\ > a}.

( i i ) The distribution function,/ . , ^ ( a ) = iJ,{E^j).

If fi i s L eb es g u e m eas u re , we r e fe r t o / . , ^ a s / .

T h e o r e m 2 . 1L e t (M , n , / 2 ) b e a g i v en m ea s u re s p ac e , w i t h / an d g m e a s u r a b l e f u n c t i o n s

def ined on M. The d i s t r ibu t ion func t ions , / . , ^ and ^ , , ^ , sa t i s fy the fo l lowingp r o p e r t i e s :

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8( a ) / , ^ is n o n i n c r e a s i n g a n d r ig h t c o n t i n u o u s .(b ) If l / l < |5r|, th e n / . . ^ ( a ) < g.,^{a), Va > 0 .( c ) S u p p o s e t h a t {fm} i s a s eq u en ce o f m eas u rab l e fu n c t i o n s w i t h t h e p ro p e r t y

t h a t \fm\ < l / m + i l , " ^ = 1 , 2 , 3 , . . . I f / i s a m eas u ra b l e fu n c t i o n s a t is fy i n g\f{x)\ = l imnôo \fm{x)\ V G M , th en for ea ch a > 0 , fm^{a) i n c r e a s e sm o n o t o n i c a l l y t o / - ( a ) a s m t e n d s t o i n f i n i t y .

( d ) { E ! I I / } . , M ( ) < E . 1 I ( / ) . , M ( " / ^ ) , V a > 0.

Proof:

( a ) A s s u m e t h a t a^ an d a^ > 0 an d t h a t a^ < a^. I t i s c lear tha t E^^j C E^^j',w h e r e b y ,

/ - , ^ (0 = 2 ) = M â2 , / ) < M â i , / ) = / . , M ( I ) -T h i s i m p l i e s t h a t / ^ ,^ is a n o n i n c re a s i n g fu n c t i o n . L e t { a } b e a n o n i n -c rea s i n g s eq u en ce o f r ea l n u m b er s wh i ch co n v e rg e t o a. T h en { ' a} n = i ,i s a m o n o t o n i ca l l y i n c rea s i n g s eq u en ce o f s e t s fo r wh i ch 0 0

U ^ ' = 'n ,/ = ^ a , jn = lI t foUows that f^^^{a)= fi{Eacj) = H m ^ _ o o / ^ ( ^ a n , / ) = l in in-00/- , / . (0=^) ,wh i ch p ro v es t h e r i g h t co n t i n u i t y o f / (a ) .

( b ) A s s u m e t h a t Vx G M , | / ( a ; ) | < 1^(2;)|- Let a > 0 be giv en. Since E^j CEa^g, the resu l t fo l lows f rom the def in i t ion o f^^^{a).

( c ) L e t a > 0 b e g i v en . F r o m t h e a s s u m p t i o n s ,^ / n , a = {x : \fn{x)\ >a}C Ef^^,,^ C ... C EF,^-

h f o U o w s f r o m t h e M o n o t o n e C o n v e r g e n c e T h e o r e m t h a tlim^^^ j XE^^Â^WW = J Ê,Jt)dfl{t) = fM{EF ,^),

wh i ch g i v es t h e r e s u l t .

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( d ) A s s u m e t h a t | E I i / d > c^. T h e n | / , | > a/N for some i, 1 < i < N.H o w e v e r, t h is i m p li es t h a t ^ ^ ^ ^ ^^ C U ^ i E^/Njr The conclus ion fo l lowsi m m e d i a t e l y .

R e m a r k 2 . 1As s u m e t h a t / i s a m eas u rab l e fu n c t i o n , f i n i t e a . e . S i n ce / (a ) i s t ak en wi t h

re s p e c t t o L eb es g u e m ea s u re , i t i s a l s o ev i d en t t h a t / - ( a ) < o o , Va > 0 , if an donly if l ima-.oo f{c) = 0.

D e f i n i t i o n 2 . 2 ( n o n i n c r e a s i n g r e a r r a n g e m e n t )L e t (M , n , / x ) b e a g i v en m eas u re s p ace , w i t h / a m eas u rab l e fu n c t i o n d e f i n ed

on M . For al l < > 0 , define th e nonincreasing rearrangement of / asf;{t) = inf{a : /.,^(a) < t} .

T h e o r e m 2. 2 ( P r o p e r t i e s of / a n d / " )Let / and g b e m eas u ra b l e fu n c t i o n s d e fi ned - o n a g i v en m ea s u re s p ace

( M , 7 , / i ) . T h e n o n i n c r e a s i n g r e a r r a n g e m e n t s of / a n d g sat isfy the foUowingp r o p e r t i e s :

( a ) f;{t) is m o n o t o n i c a l l y d e c r e a s i n g .(b ) For each e such th a t 0 < e < / . . ^ ( a ) , / ^ ( / . . ^ ( o : ) )

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10(f) {Convergence Property): Su p p o s e t h a t { / ^} is a s eq u en ce of m ea s u r ab l e

f u n c t i o n s w i t h t h e p r o p e r t y t h a t Vx G M , \fm\ < | / m + i l , m = 1 , 2 , 3 , . . . I f/ i s a m eas u rab l e fu n c t i o n s a t i s fy i n g | / ( 2 ; ) | = U m^^ oo 1 /^(^)1 , Vx G M .t h en fo r each t > 0, f^^^{t) i n c rea s e s p o i n t wi s e , m o n o t o n i ca i l y t o f ' ^ t ) .

T he p roofs o f the se a re g iven in [6 , pag es 161 th ro ug h 166] . W e om i t thed e t a i l s .

T h e re i s a s i g n i f i can t r e l a t i o n b e t ween a fu n c t i o n an d i t s n o n i n c rea s i n g r ea r -r a n g e m e n t . I t m a y b e s t b e u n d e r s t o o d f r om T h e o r e m 2 . 3 , w h i c h is q u o t e d w i t h -ou t pro of (see [6 , pa ge 16 2]) . In the fo l lowing di scu ssio n, we define R ^ = (0 , oo ) .

T h e o r e m 2 . 3L e t (M , Q , / ) b e a c r- fin it e m e as u re s p ace . S u p p o s e $ : R " ^ > R "^ i s a s t r i c t l y

i n c rea s i n g , r i g h t co n t i n u o u s fu n c t i o n , s a t i s fy i n g ^" ^ (R ' ' ' ) = R "^ . T h en fo r an ym e a s u r a b l e f u n c t i o n / : M - R ,

lMm{x)\)dfM{x) = J^^'{a)f.,,{a)da= -!^^x)df^Jx).

I n t u i t i v e l y , t h i s s u g g e s t s t h a t t h e n o n i n c r e a s i n g r e a r r a n g e m e n t r e o r d e r s t h eran g e o f / s u ch t h a t / ^ i s n o n i n c rea s i n g an d t h e a rea b en ea t h t h e cu rv e f;{t) c o r r e s p o n d st o t h e a r ea b en ea t h / o v e r i n t e rv a l s o f eq u a l m eas u re s i n t h e d o m a i n s o f / ^ an d/ . T h i s i s i l l u s t r a t ed g rap h i ca l l y in t h e ex am p l e s g iv en i n C h a p t e r I I . C o n s i d e rt h e f o l l o w i n g i m p o r t a n t i m p l i c a t i o n o f T h e o r e m 2 . 3 .

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11C o ro Uary 2 . 1 (E q u a l i t y o f t h e Z p , , No rm s o f / an d / " )

L e t / b e a m eas u rab l e fu n c t i o n d e f i n ed o n t h e m eas u re s p ace (M , fl,fi). T h e nVp, 1 < p < oo ,

I 1 / I Ip I l /pi I I p -

Proof:As s u m e t h a t p < o o . B y T h eo re m 2. 2 / an d / . a r e eq u i m eas u ra b l e fu n c t i o n s .

T h e co ro l l a ry fo ll ows a s an i m m ed i a t e co n s eq u en ce of T h eo re m 2 . 3 , w i t h # (x ) =x^. S u p p o s e t h a t p = o o . S i n c e

/ ; ( 0 ) = ii/iiooan d / J i s m o n o t o n i ca l l y d ec rea s i n g , t h e r e s u l t fo l l o ws i m m ed i a t e l y .


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122 . 2 E x a m p l e s

W e p ro ceed t o g i v e ex am p l e s o f t h e d i s t r i b u t i o n fu n c t i o n s an d n o n i n c rea s i n grea r r an g em en t s o f ce r t a i r i m eas u rab l e fu n c t i o n s . W e wi l l r e f e r t o t h e s e ex am p l e st h r o u g h o u t t h e t h e s i s .

E x a m p l e 2 . 1L e t (M , n , / ) b e an a r b i t r a ry m eas u re s p ace , an d A a s e t of f in ite / i -m eas u r e .

L et 1, X e A,f{x)=Xp^{x)=2.


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f ( t )

f,Wu(H)

f(x)

u(fl)

13

H X

Figure 2 . 1 : {X)=:A, ^OC), r{t)


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14It fol lows from Example 2.1 that

while f-{t) = g'{i) = 0, V 1. Therefore, for i (1,2) ,i r ( ) + S-(i)l = 0,

while( / + s )" ( ) = i -

We proceed to give exam ples of selected funct ions, their dist r ibu t ion funct ions,^and nonincreas ing rearrangements .

Example 2 .3Let M = { : |x| > 0} and /x be Lebesgue measure on M. For r a fixed Realnu m be r, r > 0, define / , . : M ^ R by fr{x) = | x | " ^ / ^ T h e n

Ea.,r = {x: \x\

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15Example 2 .4

Consider the measure space {M,Q.,fi), where dfi{x) = x^/^"^ M = [0 .1) , and/3 > 0. T he n /x is a r-finite m eas ur e on [0,1 ). Let f(x) = x~^, 0 < 7 < 1. Wehave

f.,,{a) = /x({x : | / ( x ) | > a})= fi{{x G M : x < a-^/^})

Clearly,

Jf"\'^^-'dx, a>l,jy^-'dx, a < 1,(3a^0 a>l.

a (3,{lY' 0 0. By reasoning

/ , . ( a ) = | ( { x > 0 : | / ( x ) | > a } |= |{ x : 0 < X < a - " } |

= /0 dx' a-^l\ a>0.

= 0, fr>.{a) t-\ T h e r e f o r e ,

f:{t)=inf{a'.fr.{a) 0.( T h e r e l a t i o n s h i p b e t w e e n /^, /,.,, and / ; is i Uu s t r a t ed g rap h i caUy in F i g u r e 2.2,p a g e 17.)

No t i ce a l s o in t h i s e x a m p l e t h a t /^ is its own n o n i n c r e a s i n g r e a r r a n g e m e n t .M o r e g e n e r a l l y , we now s h o w t h a t any n o n n eg a t i v e r i g h t co n t i n u o u s , m o n o t o n ed e c r e a s i n g f u n c t i o n w i t h s u p p o r t on (0 ,oo) is its own n o n i n c r e a s i n g r e a r r a n g e -m e n t w i t h r e s p e c t to L e b e s g u e m e a s u r e .

P r o p o s i t i o n 2.1L e t / be a m o n o t o n i caUy d ec rea s i n g n o n n eg a t i v e m eas u rab l e fu n c t i o n d e f i n ed

oiL y C [ 0 , o o ) w i t h L e b e s g u e m e a s u r e . T h e n / (x) = f'{x) a l m o s t e v e r y w h e r eo n 3 ^ .

Proof:V a > 0, {x : | / ( x ) | > a } = {x : x < f'^{a)}. It fo l lows that f.{a)= / " ' ( a ) .

C o n s e q u e n t l y ,r{t) =inf{a'.f-'{a) f{t)}= f{t), V 0 .

T h i s c o n c l u d e s the proof.


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f ( t )

{ '

Figure 2 . 2 : f ( x ) = x - ^ r > 0, f , (a ) = a"^ /^ r{t) = f

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182 . 3 T h e L o r e n t z S p a c e s a n d I n t e r p o l a t i o n

W e n o w d e f i n e t h e L o ren t z s p aces , Lp^q, s t a t e t h e i r m a i n p r o p e r t i e s , a n d p r e -s e n t i n g r e l e v a n t e x a m p l e s . W e a s s u m e t h a t f : M ^ R is a L e b e s g u e m e a s u r a b l efun c t io n . W e be g in th e d i scu ss ion by def in ing the Lp^q norm, f rom which wewiU ou t l ine the charac te r i s t i cs o f Lp , , . The p roofs fo r the comple teness o f theLpq s p aces an d o t h e r p ro p r t i e s t o b e i n t ro d u ced a re p re s en t ed i n [7 , C h ap t e r V ,Sec t ion 3 ] an d in [6 , pa ges 157 - 169] . Ex ce rp t s f rom thes e work s a re ind ica ted .W e o m i t t h e d e t a i l s .

Def in i t ion 2 .3 (||/||p,q)For eac h p ,q such th a t 1 oaf.,{a) / p

R e m a r k 2 . 2I t m a y b e s h o w n t h a t

supt>ot'^T{t) = sup^^oaf.{a)^^^,(refer to [7 , pages 192 - 195]) .

Def in i t ion 2 .4 (Lp , , )L e t / b e a m ea s u r ab l e fu n c t i o n d e f in ed o n (M , Q , / x ) . T h en / l ie s i n t h e s e tLp,q{M,n,fi), l


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25/77

20T h e o r e m 2 . 5 ( C o m p l e t e n e s s o f L p , )

For 1 < p < oo an d 1 < 5 < cx), th e Lore n tz Spa ce ( ip , , , | | . | | ; , , ) is a B an ac hs p a c e .

R e m a r k 2 . 3T h e e q u i v a l e n c e b e t w e e n | | . | |p , , an d | | . | | ; ^ s ta te d in Th eo re m 2.4 faUs fo rp = 1

a n d q, 1 < q < oo. Si n ce / " (^ ) i s m o n o t o n i ca l l y i n c rea s i n g an d r i g h t co n t i n u o u s ,i t i s c lear tha t

\ \ m . , = i ^ i f o n x ) d K x ) ] ' ' j ,which i s f in i t e i f and on ly i f f=0 a lmos t everywhere .

In [7, p a g e 2 05 ], S te in sh ow s t h a t for p = l , l < g < o o , Lp^q c a n n o t b en o r m e d . I t m a y b e s h o w n t h a t f o r p = q = loTp=l,q = oo, Lp^q i s co m p l e t eb u t Vp , g < 1 , t h e s p aces a r e n o t n o rm ab l e . Fu r t h e r m o r e , it m ay b e s h o wn t h a tfor all p > 1, the Z 'p ,oo spaces are quasi-normed under | | . | |p ,oo. In part icular ,||.|!p,oo satisfies

| | / + ^ | | p , o o < 2 ( | | / | U + | | ^ | U ) .( see [6, pa ge s 166 - 168] fo r th e p roof o f the se res u l t s ) . W e therefo re re s t r i c t ou rdiscussion to the Lp,q spaces, 1 1, q= oo .

We c la r i fy these resu l t s by the fo l lowing examples . We beg in by p resen t inga f am i l i a r o p e ra t o r wh i ch wi U b e r ev i ewed i n C h ap t e r I I I a s an i d ea l i l l u s t r a t i o no f t h e ap p I i cab i Ht y o f t h e Gen e ra l i zed Yo u n g ' s In eq u aUt y .

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21E x a m p l e 2 . 6

Def i n e t h e co n j u g a t e Po i s s o n Kern e l ,^ , , sin(x)^(^) = i - h ^ - n < x < n .1 C 0 5 ( x j

C o n s i d e r t h e co n j u g a t e fu n c t i o n / ' ^ (x ) = Q * f{x). W e s h o w t h a t Q{x) Ues in^p,oo Vp > 1 , but that Q ^ L^. It foUows that Q Ues in no Lpl-U,Il] s p a c e ,1 | ^ |, V^ 6 ( - n / 2 , n / 2 ) , it fo llo ws t h a tg . ( a ) < ( l / ^ ) . ( a ) = - ,a

w h e r e b y| | Q l l p , o c = 5 u p , > o ag ; / P (a ) < sup^ô2'/â'-'/r

In p a r t i cu l a r , | | Q | | i , o o . < 2^^^. C o n s e q u e n t l y Q G iî.oo-

E x a m p l e 2 . 7Let / ( x ) = |x | -^ /^ , fo r | x | > 0 . I t i s c lear f rom Ex am ple 2 .3 th a t Vp , 1


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22E x a m p l e 2 . 8

For |x | > 1 , define / i , (x) = \x\-^l% for f ixed r > 0 . By an easy calculat iona n a l o g o u s t o t h a t p r e s e n t e d i n E x a m p l e 2 . 3 , p a g e 1 4 ,

K{t) = {t/2 + i)-^i\ te{o,oo).I t therefo re fo l lows tha t

r= {2( ^ x^-^ /Mr}, (2.1)r q

\I^\., = { / H > , / , ( x ) ' i } ' /

= { / | x | > i i a : i - ' " x } V ^, 1 - q / r j oo

w hic h is f in ite if a nd o nly if 5 > r . Ho we ver,\\hr\\,,, = J^[t'/^h;{t)]'^^

= J^t^l^-\t/2 + l)-^l^dt= 2 / p / ^ ( t - i ) 9 / p - i r ' / ^ ( i p2 ^ 1. Let E C R be a Lebesgue measurable set of f ini te measure.

T h e n\\XE\\,,., = {i7^'"^[^ki=^)]'Ty"

while

F u r t h e r m o r e , the ra t io

= {/If'x^/P^-Mx}^/= {p2/q)''W^ ,

\ E\\p.,, = {Pi/q)' '\E\''''

EWPUQ_ Pl"^ | ^ | ( l , / p i - l / P 2 )'^ ' l lp2,9 \P2

is clearly not b o u n d e d as a funct ion of |i?|, as \E\ -^ 0. Similarly, the ra t ioM # i ^ is not b o u n d e d as \E\ -^ oo. It\\'^S\\pi,qo inclusion for the spaces Lp.^q, z = 0 , 1 .Il-^^lip'" is not b o u n d e d as |J5| > oo. It follows that we can make no s t a t ement I ' * ' B I P I , 9

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25The p roof o f the foUowing resu l t may be found in i7 , page 195] . I t impl ies

t h a t i n o r d e r t o d e m o n s t r a t e t h a t a n o p e r a t o r T is b o u n d e d w h e n c o n s i d e re da s a n o p e r a t o r o n Lr,i, i t i s su f f ic ien t to demons t ra te tha t i t i s bounded on there s t r i c t i o n o f i t s d o m a i n t o t h e c l a s s o f ch a rac t e r i s t i c fu n c t i o n s o f m eas u rab l e s e t so f f i n i t e L eb es g u e m eas u re . I t s co n c l u s i o n m o t i v a t e s t h e d ev e o p m en t p re s en t edi n C h a p t e r I I I .

T h e o r e m 2 . 7Su p p o s e T i s a l i n ea r o p e ra t o r wh i ch m ap s t h e f i n i t e Un ea r co m b i n a t i o n s o f

c h a r a c t e r i s t i c f u n c t i o n s !E of L eb es g u e m eas u rab l e s e t s E o f finite measurei n t o a v ec t o r s p ace B wh i ch i s en d o wed wi t h an o rd e r p re s e rv i n g n o rm | | . | | . I f

| r ^ , | | < C | | ^ ; | k l ,t h e n t h e r e e x i s t s a c o n s t a n t A s u c h t h a t

i i r / i i < A | i / i i , for all f i n t h e d o m a i n o f T .

In t h e proof, (which may be found in [7 , page 194] ) , the au thors make use o ft h e fo Uo wi n g u s e fu l r ea r r an g em en t o f an a rb i t r a ry s i m p l e fu n c t i o n . S i n ce we wi Uuse th is in the f inal chapter of th is thes is , we present i t br ief ly in the foUowingl e m m a .

L e m m a 2 . 1 ( A r e a r r a n g e m e n t o f s i m p l e f u n c t i o n s )L e t fN = E l i CPCE,, wh ere V , l... > CN. Defme CN+I = 0. F o r l

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26

where b^ = c^ - Ck+i, 1 < k < N. Defining

, = (f ,) S=,M.), ;>o,1 0, j = 0,it is shown in [7] tha t

fN^{oc) = < U, Ci < a,

" ^ ^ 0 , t> d^,an d Vp, 1 < p < oo,

NII/ATIIP.I = PX ^& JI I ' ^F J I P , I .J = l

We now define the concept of restricted weak type operators on Lp^q andpresent the Marcinkiewicz Interpolation Theorem for subadditive operators onLp^q. We shall rely heavily on these results in the proof of the extension of Young'sInequaUty given in th e final ch ap te r. W e assume tha t T : Lp{M,fi) -^ M{N) isa sublinear operator.

Definition 2.6 (Operator of Type (p,q) on Lq)T is of type (p,q) if there exists a constant B, such that

| | r / | | , < ^ l i^ iip,^feLp{M,fi).

Definition 2.7 (Operator of Weak Type (p,q) on Xp(/x))T is of weak type (p,q) if there exists a constant A such that

{Tf)M 0.


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27

E x a m p l e 2 . 1 0L e t / G i p ( R " ) . D e fin e P,{t) = j i ^ - I t i s weU known tha t the Po issoni n t e g r a l o p e r a t o r , Tf{t) = sup.^oP. - f{t), i s the so lu t ion o f a Di r ich le t p rob lem.(A development o f the bas ic theory o f Harmonic func t ions may be found in [6 j . )It is shown in [7] that T is of weak type (1 ,1 ) . We show tha t T i s no t type ( 1 , 1 ) .

L e t / (x ) = ^ (o , i ) (x ) . T h en fo r each x > 1 ,r ( / ) ( x ) = suptyoJ:^Pt{s)f{x - s)ds

> Yi^y'P{{x-i)

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28Theorem 2 .8

Suppose tha t T i s a subaddi t ive opera tor of res t r ic ted weak types ( r^ .pi ) ,i = 0 , l , an d th a t 7*0 < 7*i, po / Pi- T h en t h er e exists a co ns tan t Kt, i ndependentof / , such that Vg, 1 < g < 00,

iiT/iip,, < /f , i i / | i ,for all / in the domain of T and to L. whenever - = , - = + , a n d

' ^^'5 p p o P l ' "0 I"! '0 < < 1.

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C H A P T E R I I IA N E X T E N S I O N O F Y O U N G ' S I N E Q U A L I T Y F O R

C O N V O L U T I O N S O F F U N C T I O N S I NT H E L O R E N T Z S P A C E S

In the present chapter we establ ish the fol lowing extension of Young's In-equa l i ty .Let / and g be m eas ura ble funct ions defined on R and assum e th at 1 < g o{oi : a A ' ^ / " ( a ) }

SUPcc>0 l .

= Up/".Case III : u = V = ooI t f o l l o ws f ro m R em ark 2 . 8 t h a t

lA' I | oo ,oo = W X -4 oc 1.T h i s e n d s t h e proof.

Before p roceed ing to ver i fy Inequal i ty 3 .2 fo r convo lu t ions o f charac te r i s t i cfu n c t i o n s o f b o u n d ed i n t e rv a l s , we ex p l i c i t l y ca l cu l a t e t h e g en e ra l fo rm s o f t h ed i s t r i b u t i o n f u n c t i o n a n d n o n i n c r e a s i n g r e a r r a n g e m e n t o f s u c h f u n c t io n s .

L e m m a 3 . 2Let A = (xi ,X2), B = (2/1 ,2/2) he in tervals on the real Une, with \A\ < \B,,T h e n \A\ + | B | - 2 a , 0 < a < \A\,{^^*A:B).{a)={

0, a > \A\.

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32Also, 1 1, 0 \A\. For a < \A\ we

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33have

{^A * A'B ) . (a ) = \{t : {XA * A'B)(t) > a } |= \{t:{XA^XB){t-x^^y^)>a}\,

( this fol lows from the t ranslat ion invariance of Lebesgue measure) , from which{XA * B ) - ( a ) = \{{t [0, |A |) : < > a } | + \{t : |.4| < t < \B\}\

+ \{t e [\B\, \A\ + \B W \A\ ^\B\-t> a}\= {\A\ - a) + {\B\ - \A\) + (|.41 ^\B\-a- \B\)

= \A\^ \B\ -2a.Thi s p roves t ha t

(3.4)

(3.5)

| A | T - \B\ -2a, 0 1.41

To ca lcula te {X ^ * P^By{t)i we consider the following three cases, (see Figure3.1 , page 34) .

(i) 0 < < < \B\ - \A\.H e r e , {X A * P^B)(^) < t if and only if a > | .4| . Consequently,

m / { a : ( ^ A * ' - ^ B ) . < ( ) < 0 = l ^(ii) \B\- \A\


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34

r( i )

ifli, IBI-Ifll iBkl fll

fWiBKIfll

IBI-Ifll"Ifll

M

i f l i - y. Xxi * gi 2*^ 1 xj+g^ ^t^Z X

Figure 3 .1: f (x)= ^ ,4 * ' ^ s , A = (x i^xj ) . B = {y^yi)

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35Since {X A * XB).^-^^ - l) = t an d {X A * B ) .( a ) is decreasing , i t followsthat

( ' ^ A * ' ^ 5 ) x ( a ) < .I t i s a l so immedia te tha t

inf{a : XA * ?CB)^{OL) Ul + I^l t(iU) t > \A\^\B\

Not ing t ha t {X A * ^f ) (a) < \A \ + \B \ for a// a > 0, it foUows that

The re fore ,

Summar i z ing t h i s ,

( '^ ^ * '^ s )x ( a ) < ^ V a > 0 .

inf{a :{XA^XB)Xa)


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37M u l t i p l y i n g b y a^ /"6 '' /' ' an d u s i n g t h e r e l a t i o n 1 / p = 1 / u ^- 1 / r - 1 , we o b t a i n

- a'^^b - af'^ + ^ ^ l ! ^ / U ( l - tyt'^'^-Ut < Ka'^'^'b'^'^ (3.8)q 2^^ '>+" ~

By L e m m a 3 .1 , pag e 30 , th i s es tab l i she s the des i red resu l t fo r finit e p and q .Case II : q = oo, and 1 < p < oo.F r o m L e m m a 3 .2 , p a g e 3 1 , a n d R e m a r k 2 .2 p a g e 1 8 ,

11'^^ * '^BIIP^OO = SUpt^of^^XA * ^B^'^t)t^'Pa, 0 l | | u , ; | | ' < B | | r , 5

s ince a < 6 an d 1/u - 1 < 0 . B y L em m a 3 . 1 , t h i s e s tab Us h e s t h e r e s u l t .

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38Case III : q = p = cx).RecaU from Remark 2.4 that | | / | |oo,oo = | | / | |oo. In this case, the condition1/p = 1/u + 1 /r - 1 bec om es 1/u ^ 1/r = 1. By definition,

I I ' ^ ^ * ' ^ B | | O O = SUP,^OJ!^XA{X - t)XB{t)dt< {f-.o^A''{x - t)dt}''-{jzxut)dty'^= a^'%^''

= | | ' ^ ^ | | u , l | | ' ^ B | | r , l , ( 3 . 9 )val id Vu, r , 1 < u , r < oo, by Ho lder 's Inequal i ty. If for instanc e, u = oo an dr = 1, the resul t is obvious.

Before proving Lemma 3.4, we present the fol lowing representat ion theoremfor finite coUections of bounded intervals.

/r

Lemma 3 .3Let e > 0 be given. Let c be an arbi t r ary posi t ive cons tan t . LetMJ=[j{a.,bi).

1 = 1

T h e n J has a representat ion of the form J = Ufcli Jk , where for each fc, J^ is abounded i n t e rva l and

( i ) E f= i ( \Jk\ - \Jk+x\ )M^ IJ2I > |^3| > > I^^MI,

( ii i) E f= i ( iAi - /-.I )A f ' < ^ -

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39Proof:

C h o o s e > 0 . L e t J i b e a s u b i n t e rv a l o f J of m a x i m u m l e n g t h . P a r t i t i o nJ i i n t o M i - 1 s u b i n t e r v a l s { J ' i , ^ } , ^ \ " ^ of e q u a l l e n g t h ^ , a n d a r e m a i n d e r H^,w h e r e M ^ i s chosen to be the smaUes t pos i t ive in teger such tha t

- ^ < ( M i M ) - ' = - 2 'M i - ' ' ' ^M 'I t f ol lo w s t h a t t h a t | 7 ^ i | < ( M i A I ) - ' ^ - ^ ^ . H a v i n g d o n e t h i s , d i v id e e a c h s u b in -te r va l J^ i n to Mfc - 1 in te rv a l s ,

M * - lJk= \^ {ji,k,ji+i,k) Ui?fci = ls u c h t h a t

( i j Ji+i,k - 3i,k < jf^,( n ) Mk < Mi, a n d

(iU) | 7 ^ , | < ^ , l < f c < A ^ .L e t Li b e t h e s u b s c r i p t , k , o f a r em a i n d e r o f l a rg es t l en g t h . T h e rea f t e r , o rd e rt h e r e m a i n d e r s , T ^ L , , 1 < * < M , i n an y o rd e r s u ch t h a t

We def ine

and for g iven t , define

I ^ L . | > | 7 ^ L , , ,

M = M i A ^(Tt = J^ML,i= l


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40Let

^ " l , L i = ( l , L i , 2 , L i + 2 - ^ | 7 ^ L i | ) ,^--2,^1 = (2,Li + 2 - ^ | 7 ^ L i b 3 , L i + ( 2 - ^ + 2 - 2 ) | 7 ^ L , | ) ,

^''Af,,-!,^! = ( M. , - i ,L i + ( E ^ ' I ^ ' ' 2 - 0 | 7 1 L I I , M , , , L I ^ ( E . ' ^ " ' 2 - ) | 7 ^ L i l ) ,(3.10)an d M i , , - 1J^Mi,^,L, =R^L, ={JML,,L, +( Y. 2- ' ) |7^Li l ,6 i ) .i= l

Thereafter , for f ixed k,J-I.L, = {jl.LJ2.L, + ^ ' ^ ' - ' ^ - N T ^ L J ) ,

J-n.L, = Un.L, + iX^^''^"'' 2-^)\Tl,,\, jn,,.L. + ( E r i T " ^ " 2 - ) i 7 e , J ) ,(3.11)

an dJ-M. .L, = R-L, = Uu.r, + ( 2-)iKiJ,aiJ. (3.12)

i = l


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For ease of notat ion, we rename these intervals as foUows. Define41

Jl - J'l^L^, (3.13)

JML-1 / ' ' M - 1 , L I ,

J, J\,L^, if i / 0,C T f c - l + lNote tha t i f i =

an d

R-L,, iii=0.

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42There fo re ,

(i) Sup pos e tha t ne i ther J^ nor J^+i are rem ainde rs. Th at is , nei ther i + 1 nori is equal to E=:i Mj OT s^ny k, 1 < k < M. It foUows thatJ\Jr \ - |Ji + i | I < |i^j < 2 -^- ^M-i-l A/f-c-2 eM i 6M '

(U) while if Ji = Rl for some k, 1 < k < M, (equivalently, if i = Z^i ^j^ ^orsome k,l < k < M), then cer ta in ly\Jr\-\Ji+l\ | < | T T ^ | < M - = - 2 eM i 6M '

It follows thatE l l ' / i |- |/ i+l | l ^ ' < E I l '/^l- l^r+ll |M=^ = 1 Ji+i,JijRl,Vk,i

+ E I l^d-|^.+il \M % (3.16)i,k3Ji = R l O ^ J ' . + I = ;From the Representat ions 3.11 through 3.12, Inequali ty 3.16 is

< E^Mvi2-^-^ |7^Li|M'^ + E ^ rM k ' i ,^+i l \M''+ E.^rM I^"M.-I,.I IM'

By the statements foUowing Expression 3.15, this implies thatE l |J . - |- | Ji+ i | ^M''

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I t therefore fol lows thatM Mj-l Me.__._^E E i l ^ \ . J - l A . l \M' I-R'L^I > I^ 'L S I > > l-R' LM \I

43

since if J ^ ' L ^ I = |-R'"Lfcl ^ ^ so m e n ,k , 1 < n,; < M, we may assign to eachJML ,Lk ^ subinterval of R^Lk ^^ length less than 2'^ -' ' |i?Lfc|. The proof thenproceeds by an argument nearly ident ical to that in Expressions 3.10 through3.12.

It follows that the intervals defined in Representation 3.13 satisfy the hy-potheses |Ji | > IJ^j > IJ3I > .. . > | J A | . In ^he following lemma, we prove thevalidity of Lemma 3.2 for characteristic functions of finite unions of intervals.

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44L e m m a 3 . 4

Let / = U n=i ^n an d J = U ^= i Jm wh ere { / } an d { J ^} a re t wo co Uec t io f d i s j o i n t b o u n d e d i n t e r v a l s . \/ t > 0 , d e f i n e / ( t ) = X^ ,N , * X, ,M (t ) Wio ns,Veassume thatThen

l^il > I/2I > ... > IJM

0,

|i 1 +- iv i^ii - -^ ,a > M | J i | ,

MIJ2I < a < MjJi l ,/ . ( a ) < { 2 |/ | + iV(|Ji) + IJ2I) - g , M jJsl

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45Proof:T h e p roof p roc ee ds by in du ct io n on N. T he case M = N = 1 foUows f rom

L em m a 3 . 2 . W e p ro v e t h e ca s e for M = 1 an d a rb i t r a r y N . As s u m e t h a t t h es t a t em en t i s t ru e fo r N , t h a t i s ;

^ ' ^ U n= l ^ ~ ^ ' ^ j } - ^ ^ ) < | J | .

B y T h e o r e m 2 . 1 ( d ) ,{^^;v.i ^^ * Xj}.{a) < {X, ,N . Xj).{a/2) + {Xi,^^ . ^j).{a/2)

 0,

û^.Vi, * '^t) = \{t - ufî/z) n J| | J | ,

('û-t^/. * '^^)''(") = 0-T h i s c o m p l e t e s t h e p r o o f f or M = l a n d a r b i t r a r y N .

No w co n s i d e r t h e h y p o t h es i s o f t h e l em m a fo r a rb i t r a ry M an d N . L e t/ = (ÛL ' .*Û "=.^ ''

a n dfi = X, ,N , *Xj., Vi , 1 < z ' < M .'* U n = l^ n ^' ' - -

B y t h e p recee d i n g r e s u l t for M = l an d a rb i t r a ry N , i t is c l ea r t h a t Vm .N,

1

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46

/ m ( a ) A / | J i | ,M I J2 I < a < M | J i | ,MIJ3I < a < MIJ2' , (3.18)

^ | / | + ^ ( E L | /i |) - ^ , M | J ^ + i | < a < M | J ; , |^ M\I\ + N{Zf=i \Ji\) - 2 a , a < M | J M |

From Theorem 2.1(e) and Inequal i ty 3.17, i t i s c lear thatM M a/ - (") = ( E / . ) - ( ) < E / i - ( T 7 ) < G ( a ) .i = l i=l ^^^

This establ ishes the dist r ibut ion funct ion of the convolut ion of characteris t icfunct ions of f ini te unions of bounded intervals .

Sincer{t) =inf{a:f.{a) ,M , )=


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4"

ZlII+ZUikZlJ^.

2111+21^"

| Ik2 |J l -2 |J j1 2I I L .2 I J 2 I 2 I J 1 I

f( )

* i a -^ m b + l b + m dVc - i ' k a-^I d-^k c - ^ I c - ^ m ^*^

F i g u r e 3 .2 : f ( x ) = ^ ^ * ^BJI = ( 0 , ^ ) ^ 2 = {c,d)/ 1 = {j-k),J2 = {Lm), / = /1 U/2


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48

2!J,

2 I J 2 I -

|I|*2(|JiJ-li!)2|Ik2(|JiklJtl)

Figure 3.3: A Bound Fo r r{t)


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49W e p ro ceed t o ex t en d t h e co n c l u s i o n o f L em m a 3 . 2 fo r t h e co n v o l u t i o n s o f

ch a rac t e r i s t i c fu n c t i o n s o f f i n i t e u n i o n s o f i n t e rv a l s .

T h e o r e m 3 . 2L e t / = U^^ i / n , J = ^m=iJm, where { /} and { J^ } a re co l lec t ions o f

b o u n d e d i n t e r v a l s . T h e r e e x i s t s a c o n s t a n t C , i n d e p e n d e n t o f I a n d J, s u ch t h a t\^i^^j\\v,,

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50(i) 1 > | / i | > I/2I > IJ3I > ... > | J M | ,

( i i ) Ef=i | | / i t | - |Jfc+i||M^+^/P+i < m m { e , e | / | } ,( i i i ) Ef=i lUil - | J , | |M^+^/P-^ < 6, where 6 = mm^.J^} ( - ^ Y '(iv) M > 7V > 4.

For each K, 1 < K < M, we adopt the foUowing notat ions:n ^ = / r | / | + 7 V E = i | J m | - 2 i : | j ; , l ,^K = i ^ | / | + N Zm = l \Jm\ - 2 / r | J ^ + i j ,

_^ K\I\ + Nj:Z_^\J^\-2K\J,\

^ Km + NZ= ,\-^rn\-2K\Jfc.l\^ i ^ i + ^ E L i i - ^ - iA = 1^2}, and /c;, = ^ ^ T i ^

By convent ion, we assign J M + I = 4>- From Lemma 3.4 and the definition of\p^q, it follows that

II V ^ ^ v ^ ||g < v-M r^ (MK\I\+NMZL.\-Jrn\-Mt.^ ,^_^W-^u^^jn* "^^^^^.J^nWp^q ^ 2^K=IJQA 2K ) i ai+(M|J i | )Vl / l+ (^ - ' ) ' ' ' ^ l ' / p -^ N - 2, w her eb y s ince ^ < 1 by ass um pt i on , we a l so haveTJ + ir Pr'^ P

2\J \ \ ^ / P - ^ 1N - ^ < ( ) ' - ^ / p .

( ^ ) 1 I '' i , l i i C < J . AN .}(^^ K .ff; N ^ K N ^ KHJ^ .I t fo l lows tha t Express ion 3 .26 i s majo r ized by

K=l ^^ -" 'A Y({\J \-\J +i\)\f 1 V ^ '

IV + iC / l N ^ K ) ^ ^ ^ ^/ 1 \ '/P

+ (p / ) A ' - ' / ' ( l + - j ,which is

< 2 X: A ^-^/^i /^+^-^n^ + -f}"-\\J \ - \J .i\) [i;^K=l q/p+ {p/q)A'-'^''-{l + \)'' (3.27)

S i n c e b y a s s u m p t i o n M > N a n d

t h e l a s t ex p re s s i o n i s m a j o r i zed b y2 ^ , - , / . ^ M ^ ^ Kq/p+q-2q ( M 1 ) ( | J ^ | _ | J ^ ^ , | ) ( ^ ) ' " ' ^ '

+(p /9)A ^-^ / ' -(l + i ) ' ^ ^Let (5 = g - g /p . Sinc e 5 > 0 by a ss um pt io n, Ine qua U ty 3 .27 is

.V- H ^ ^ ; ' ^+^'"""' " " " " l ^ ! '

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< 2

56

[-^;i''^-^"h\M-\j.^A)'i.-2^' 4 , E f= i ( | J i < : | - | J J C + I | ) M ' < e | / | , an d A < 1. It followsf ro m In eq u a l i t y 3 . 2 7 t h a t

XT * X ^W ^ n" ^ ^ " P ' g < 2 2 - 9 / ^ 6 -^ - 2 * ^ / ^| / | 9 / u | J | q / r - P 'T h i s g i v es t h e d es i r ed r e s u l t .

Case II : q/p > 1In th i s case , we fo l low an a rgument iden t ica l to tha t o f Case I, wi t h In eq u a l -

i t y 3 . 2 4 r ep l aced b y t h e i n eq u a l i t yx^ -y'' y > 0, r > 1. (3 .28)

B eg i n n i n g f ro m In eq u a l i t y 3 . 2 5 , we o b t a i n t h e r e l a t i o n

^ ^ N jc + K j \li + jcl^m=l\'^m\ (3.29)' / fc + l j ^q/p-l

NF r o m t h e a s s u m p t i o n s .

/v I- ' I f Z ^ m = l l '^ 'Til

N - i^ ^ K N A ^ K

(b ' ) M > N, a n d(c ' ) V/f , 1 < / r < M , I | J i f | - | J i f+ i | I < J MJ^ < Ar^/P-'i^M^+^iC' ^^^^^

K < M .

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57T h e r e f o r e ,

M 7 y - q / p + g - 2 ( j -, -. n TI rX, ,N . * A '|| M , jl' < Y - (KY-

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Case IV : p = q = ooF o r m e a s u r a b l e s e t s A of f n i t e m e a s u r e , we u n d e r s t a n d t h a t

; ^ 4 | l / o o _ ]_O b s e r v e t h a t

Il^-tM^ - t)û^,,r.m = K - 4UJ.) - ( u t , / ) |< r I I l ^ T \s I ^k=l -'fcl

L e t r and u be s u c h t h a t 1/r ^ I /T^ = 1, 1 < r,u < oo. It fo l lows thatIôo û^^.Jki^ - t)X^N^^j.{t)dt < UliJ.I(^^)l U f = i / f c l ^ / l U ^ i / .P /" - {ru)\ Uti Jk\"^\ U, Ir\"-

\ \M 7 l l /uUfc=:l Jk\ ' < < 1.( r u ) m ' I i / d ' / " - TuA p p l y i n g the r e s u l t s of L e m m a 3.1, th i s d i scuss ion impUes tha t

l l ' ^ u A ^ J f c * ' ^ U ^ ^ / J l o o < l l ' ^ U ^ l j j J l r . l J I ' ^ U ^ i\\u.l'

5 8

(3.30)Since th i s resu l t g ives no r e s t r i c t i o n s on the e x p o n e n t s s and v, it s eem s t h a t

i t cou ld not be o b t a i n e d f r o m an i n t e r p o l a t i o n t h e o r e m . O n l y p a r t of its s t r e n g t hi s n eed ed to p r o v e T h e o r e m 3.3. We f rst p r e s e n t the fo l lowing lemma.

L e m m a 3.5L e t f be a n o n n e g a t i v e m e a s u r a b l e f u n c t i o n and {/n} a s eq u en ce of n o n n e g a -

t i v e m eas u rab l e fu n c t i o n s s a t i s fy i n gUm /n = /,n x)

a . e ., m o n o t o n i c a l l y . T h e nUm | | /n | |p,q = 11/! P , 9 "

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59

Proof:By Theorem 2 .2( f ) ,

Um r = / -pointwise, monotonically. Consider first the case p < oo. For fixed p and q, define

hn{t) = f^'^-'f'M'-By the monotone convergence theorem, i t foUows that

lirn^ J hr,{t)dt = h{t)dt,while if p = oo,

lim^ sup^yoa^'Pf;^{a) = sup^^^a^'^^{a),by Theorem 2.2(f) . This gives the desired resul t .

Theorem 3.3 (Restric ted Weak Type (u,p) for Convolut ions on /p,q)Let p,u,r > 1 and 1/p = 1/u + 1/r 1. Then there exists a constant C,

independent of / and g, such tha t| | / * ^ | | p , o o < C | | / | k i | | ^ | k l ,

for aU / G Xu ,i, 9 G / r . i -

Proof:Let F and G be arbi t rary sets of f ini te measure. We define/ = Xf an d g = XQ.

We prove the validity of the conclusion for / and g. The extension to arbi t rarymeasurable funct ions wiU foUow from Theorem 2.8 and the context of the proof

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60g ive n in [7 , pa ge s 195 - 196] . Th e rea de r shou ld reca l l Re m ark 2 .3 , pag e 20 . Weshal l never consider the case p=oo unless also q=oo, s ince | | . l | cx3,q is not def nedo t h e r w i s e .

Let e > 0 . By [4, pa ge 62] , th er e ex i s t fin i te un io ns of bo un de d in te rv a l s ,/ = UL i / n an d J = U^^ i J ^ ,

wh i ch s a t i s fy

a n d

Definea n d

Thenan(

l ( ^ " ^ ^ ) ^ ( ^ ^ " ^ ) l < 6 m a . { 2 | C i , 2 | F ! p / ^l ( ^ ^ ^ ^ ) " ( ^ ^ ^ " ) l < ( 6 r n a x { 2 | J | , 2 | F | p / - )

u = {FnF)u {F" n /)(3 = {Gnr)u{G''nJ).

F C U ^ = i / n U 2/,

G C U ^ = l J n . U 3 .

(3 .31)

l / r + l/uA^/^R e m a r k 3 . 2I t fo ll ow s f ro m A s s e r t i o n 3 .3 1 t h a t f or e < ( 6 m m { 2 l F | , 2 | G | } ' " + ' "y| / | < m m { 2 | F | , 2 | G ! } . (3.32)

Similarly, !J i

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61

We have11/ * 9\\p,oo = \\XF * ^GIIP,, < 11^,^ , * ^JU^IIP.CX,.

B y R e m a r k 2.3 page 20, it foUows thatll-^/u. * P^Ju\\p,oo < 2^{\\Xi * XJWP^^ + IIA, * ^ ^ I I P , ^ + l l^ , * XJWP^^ ^ \\X, * XeWp,^)

Cons ide r the express ion ||;f/ * ;f^||p,oo. As in [4, page 56], let/ n _ I oo A^ ^m=l-^mi

where A^ are a sequence of bounded open inte rva ls , /? C C, and' ^ ' < V 3 ^ a { 2 | F | , 2 | G | } i / - j " ^^'^^)

For each M, de ine OM = Uf^ iA^ . Then| ' ^ / * ' ^ / 3 | | p , o o < | | ^ / * ^ o | | p , c x , . ( 3 . 3 4 )

Since Xo^ * Xi increases to XQ * XJ, it follows from the Monotone ConvergenceT h e o r e m t h a t

li m Xo.,^ * Xj = XQ * Xj.B y L e m m a 3.5,

lim ll^ o^ * '^/||p,oo = W^o * '^/llp,oo. (3.35 )M - + 0 OHowever , Theorem 3.2 impl ies tha t VM,

| | ^ 0 . . * ' ^ / | | p . o o < i ^ | | ' ^ O M l k l l l ' ^ / l k l -It foUows from Lemma 3.5 and E q u a t i o n 3.35 by taking l imits that

l lA 'c j * ' ^ / l l p . o o < / ^ l l ' ^ o l k l l l ' ^ / l l u . l

= {ur)K\I\'"^\0\^l''< ( ^ - ^ ( " ^ ) ^ | / | i / V 3 m a i { 2 ; F : , 2 ; G i } i ' V I '

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62By Asser t ion 3.32,

l l ' ^ / * ' ^ / 3 | | p , o o < | | ' A : ' / * A ' o | | p , o o < ^ ^ ^ .By s imi la r reasoning,

l l^ ' . ; II

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63T h e r e f o r e ,

l l / * ^ l l p , o o < 2 ^ | | ' V / * ^ j | | p , o o ^ e i : ( u r )< ^ ' l l ' ^ / | | u , i | | ' ^ j | | . , i + e i ( u r )< / r ' ( ^ r ) | / | i / " | J | V ^ + e i r (7 . r )< 2"-^^'^K'{ur)\F\"^\G\''^ + eK{ur).

Si n ce e is a r b i t r a r y , t h i s i m p U e s t h a tll '^F * '^G|IP,OO < / ^ l l ' ^ F l k i l l ' ^ G l l r . i + eK{ur).,

w h e r e K' = 2^''^^^'''K'.F i x B an a r b i t r a r y set of f i n i t e m eas u re . Let g = XB- Define the o p e r a t o r

'J'g ' Lr,i -^ M{N) by Tg{f) = g ^ f. For any set A of f n i t e m e a s u r e , we h av es h o w n t h a t

l| ; ( ' ^ A ) l l p , o o < / r ' | | ' ^ . i l k i l l ^ l l u , i ; ( 3. 36 )w h e r e K' is a c o n s t a n t i n d e p e n d e n t of / and g. It foUows from the proof ofT h e o r e m 2.7 in [7, p a g e 195], t h a t t h e re ex i s t s a c o n s t a n t C, i n d e p e n d e n t of /a n d g, s u c h t h a t I | ;( /) IU

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64w h e r e K" i s a co n s t an t d ep en d i n g o n l y o n p,u, an d r . T h i s co m p l e t e s t h e p ro o fo f T h e o r e m 3 . 3 .

Now le t l i be a f ixed cons tan t , 1 < u < oo, an d a s s u m e t h a t fo r i = 0 an d 1 .Pi an d r^ a r e a rb i t r a ry co n s t an t s s a t i s fy i n g

1/pi = 1/ri + 1/u - 1 , ro < r i .L e t g G / u . i . B y t h e p r e c e e d i n g d i s c u s s i o n , Tg i s an opera to r o f res t r i c ted weakt y p e s {ri,pi), w i t h t y p e c o n s t a n t s d = K'-\\g\\^,i, i = 0 , 1 . T h a t i s ,

l i r , ( / ) | | p . , o o < i ^ r i l ^ l l u , i l l / l k i , ^ = 0 , 1 .I t f o l l o ws f ro m t h e p receed i n g a rg u m en t an d t h e M arc i n k i ewi cz In t e rp o i a t i o nT h e o r e m t h a t t h e r e e x i s t s a c o n s t a n t Kt^, i n d ep e n d en t of / a n d g, s u ch t h a t

| T , ( / ) | | p , , < i ^ e | | ^ | | u . i | | / l k , V r, a n d p ,s a t i s fy i n g t h e r e l a t i o n :

l/rt = t/r, + {l-t)/r (3 .37)1/P = V P O + ( 1 -t)/pi,

for aU t , 0 < t < 1 . Since by assumption 1/pi = 1/u + 1/r, - 1, {i = 0 , 1 ) ,1/pt - 1/rt = t{l/po - 1/TO) + (1 - t ) ( l / p i - 1 / r i ) .

= t{l/u-l) + {l-t){l/u-l)= 1/u-l.

W e m us t ver ify th a t th e converse ho lds . Le t r > 1 and p > 1 sa ti s fy 1 /p =1/u + 1/r - 1 , for som e fixed u, 1 1 and pi > 1 sucht h a t 1/pi < 1/p < 1/po. (3-38)Hav i n g d o n e t h i s , we m ay ch o o s e r^ , s u ch t h a t 1/pi = I/T -f 1/u - 1, ( i = 0 . 1 ) . I tfo Uo ws t h a t 1/pi - 1/po = 1/ri - 1/ro.

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65B y R e l a t i o n 3 . 3 8 , 1/p = t/po + (1 - t)/p\, for som e t G (0 ,1 ) . He nce ,

I / T - = t/po + (1 - t ) / p i - 1/-I + 1= /P o + ( l - 0 / P i ^ l A o - l / p o= (1 - ) ( l / p i - 1 /po) + 1 / ro= ( l - 0 ( l / r - i - l / r o ) + l / r o

r i rij ro

= + llzHro r iT h ere fo re , r s a t i s f i e s C o n d i t i o n 3 . 3 7 . B y ap p l y i n g t h e M arc i n k i ewi cz In t e rp o l a -t i o n T h e o r e m t o t h e o p e r a t o r Sf : g ^ f * g, we ob ta in the foUowing resu l t .

T h eo rem 3 . 4 (An E x t en s i o n o f Yo u n g ' s In eq u a l i t y fo r / p , , )Let u , r > 1 , p > 1 , 1 < q",q' < q, and 1/p = 1/u ^ 1/r - 1. T h e n t h e r e

ex i s t s a co n s t an t K s u ch t h a t| | / * ^ l i p . , < i | | / l l u . , ' | l ^ l k , ' S ( 3 - 3 9 )

f o r a U / G / ^ u . q ' , 9 ^ - ^ r . q " .

T h i s co m p l e t e s t h e d ev e l o p m en t an d g i v es t h e p r i m ary r e s u l t o f t h e t h e s i s .

R e m a r k 3 . 3No t i ce f ro m t h e p ro o f p re s en t ed ab o v e t h a t t h e co n c l u s i o n o f T h eo rem 3 . 4 i s

no t vaUd if p = l . T h i s re f lec t s the fac t th a t an ope ra to r o f weak type (1 ,1 ) i s no tn eces s a r i l y o f s t ro n g t y p e (1 . 1 ) -

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66In c los ing we p resen t the fo l lowing example , which i s sugges t ive o f the u t iUty

o f T h e o r e m 3 . 4 .E x a m p l e 3 . 1

C o n s i d e r t h e conjugate function, / " = Q * / , of a m ea sur ab le func t ion /d e f i n ed o n t h e u n i t c i r c l e . W e h av e f ro m E x am p l e 2 . 6 , p ag e 2 1 t h a t

liQlll.oo < OO.B y a n a p p U c a t i o n o f T h e o r e m 3 . 4 ,

| | Q * / | | l . o o < ^ I I Q I | l . o o | | / l i l , l ,for s o m e co n s t an t K , i n d ep en d e n t o f / . T h i s g iv es a d i r ec t p ro o f t h a t Q * f 'isa res t r i c te d w eak t yp e (1 ,1 ) ope ra to r . ( In [7 , pag e 236] , i t i s shown th a t /= i s at y p e (2 , 2 ) o p e ra t o r . ) I t f o l l o ws t h a t f i s of type (p ,p) , Vp, 1 < p < 2 .


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REFERENCES

[1 ] J . B . Conway: Functions of One Com plex Variable, Sp r i n g e r -Ver i ag , NewYo rk , 1 9 7 8 .[2 ] P . R . Halmos : Measure Theory, D. Van No s t r an d C o . In c . T o ro n t o , 1 9 5 0 .[3 ] R . A. Hunt : "On I^p , , ) Spaces , " L'EnseignementMathematique. 12 (1966) ,p p . 249 - 275 .[4 ] H. L . Royden : Real Analysis, 2 n d E d i t i o n , M acm i Uam Pu b Us h i n g , NewYo rk , 1 9 6 8 .[5 ] W . R u d i n : Principles of Mathem atical Analysis, 3 d E d i t i o n , M a c G r a wHiU, 1976 .[6 ] C . Sadowsk i : Interpolation of Ope rators and Singular Integrals, I s t Ed i -

t i o n , M arce l Dek k e r , In c . New Yo rk , 1 9 7 9 .[7] E. Stein Sz G . W e i s s : Introduction to Fourier Analysis on Euclidean Spaces,I s t E d i t i o n , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1 9 7 1 .[8] E. Stein Sz G. W ei s s : "An E x t en s i o n o f a T h eo rem o f M arc i n k i ewi cz an d So m eo f I t s A p p l i c a t i o n s , " Journal of Mathem atics and Mech anics, 8 (1 9 5 9 ) ,p p . 263 - 284 .[9 ] D . W e i n b e rg : "L ec t u re s i n R ea l An a l y s i s . " L ec t u re s f ro m a co u r s e i n R ea lAn a l y s i s a t T ex as T ech Un i v e r s i t y , Sp r i n g , 1 9 8 6 .

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P E R M I S SI O N T O C O P YI n p r e s e n t i n g t h l s t h e s is i n p a r t i a l f u l f i ll m e n t o f t he

r e q u i r e m e n t s f o r a m a s t e r ' s d e g r e e at T e x a s T e c h U n i v e r s i t y . I a g r e et h a t t h e L i b r a r y a n d m y m a jo r d e p a r t m e n t s h a l l m a ke it f r e e l y a v a i l -a b l e f o r r e s e a r c h p u r p o s e s . P e r m i s s io n t o c o p y t h is t h es i s f o rs c h o l a r l y p u r p o s e s m a y b e g r a n te d b y t h e D i r ec t o r o f t he L i b r ar y o rm y m a jo r p r o f e s s o r . It i s u n d e r s to o d t ha t an y co p y i n g or p u b l i c a t i ono f 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 n ot b e a l l o we d wi t h o u t m yf u r t h e r wr i t t e n p e r m i s s i o n a n d t ha t a n y u s e r inay b e l i a b l e f o r c o p y -r i g h t i n f r i n g e m e n t .

D i s a g r e e (P e r m i s s i o n n o t g r a n t e d ) A g r e e (P e r mi s s io n g r a n t e d )

S t u d e n t ' s s i g n a t u r e S c u a n t ' s s i g n a t u r e

D a t e ^bsk^D a t e


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I


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lorentz spaces youngs inequality convolutions thesis

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