S. V. Astashkin- Fourier-Rademacher Coefficients of Functions in Rearrangement-Invariant Spaces

8/3/2019 S. V. Astashkin- Fourier-Rademacher Coefficients of Functions in Rearrangement-Invariant Spaces

1/10

S i b e r i a n M a t h e m a t i c a l J o u r n a l , I / bl . 41, N o. 4, 2000

F O U R I E R - R A D E M A C H E R C O E F F I C I E N T S O FF U N C T I O N S I N R E A R R A N G E M E N T - I N V A R I A N T S P A C E S

S . V . A s t a s h k i n U D C 5 17 .9 8 2.2 7

i . P r e l i m i n a r i e s a n d D e f i n i t i o n soOL e t { 'Wk }k =~ b e a u n i f o r m l y b o u n d e d o r t h o n o r m a l s y s t e m o f f u n c t i o n s o n t h e i n t e rv a l [0 , 1 ]. G i v e n

, OOa f t m c t i o n x E L I [ 0 , 1 ], d e n o t e i t s F o u r i e r c o e f f i c ie n t s w i t h r e s p e c t t o { W k } k = l b y C k = c k ( x ) .I f F i s t h e o p e r a t o r c a r r y i n g a f i m c t i o n x t o t h e s e q u e n c e o f i t s F o u r i e r c o e f fi c ie n t s , F x ----( C k ( X ) ) ~ . = l , t h e n F a c t s b o u n d e d l y f r o m L 2 i n t o 12 a n d , b y M e r c e r ' s t h e o r e m [1 , p . 1 81 ], f r o m L 1i n t o c o. T h e r e f o r e , f o r e v e r y x E L 1 w e h a v e~ , ( t , F x ; c o , 1 2) < A 1 , ) t ~ ( t , x ; L t , L2) ( t > 0 ) , (1)

w h e r e ~ z ( t , x ; ) c 0 , x ~ ) = i n f { l l z 0 L v , ) + t l [ X l L ~ I : ~ = x 0 + ~ 1 , z ~ E z ~ }i s t h e P e e t r e ~ , - f u n c ti o m ~ l d e f i n e d f o r x E X 0 X 1 ( ( X 0 . X 1 ) i s a n a r b i t r a r y B a n a c h p a i r ) , t > 0 ,a a l d A 1 = m a x { 1 , s u p { [ w k ( t ) ] : k = 1 , 2 . . . . , t e [ 0 , 1 ] } } .B y m e m ~ o f ( 1) t h e f o l lo w i n g e s t i m a t e f o r t l m F o u r i e r c o ef fi ci en t s o f a f u n c t i o n x E L l w a so b t a i n e d i n [ 2 ] :

n 2, ~1 /2 / 1 / n ( ~ , ~ 1 / 2 ]f x ' ( t ) i x ' ( , ) l ' . ?k = 1 " 0 I / n

( 2 )

w h e r e n = 1 , 2 , . . . , a n d x * ( t ) a n d ( C k ( X ) ) k = 1 a r e d e c r e a s i n g r e a r r a n g e m e n t s o f t h e f i m c t i o n I x ( u) [ a n dt h e s e q u e n c e ( k ' ~ ( ~ ) [ ) ~ = i -R e l a t i o n ( 2) c a n b e t r e a t e d a s t h e B e s s e l i n e q u a l i t y f o r s m n m a b l e f i m c ti o n s ; m o r e o v e r , i t i s e x a c ti ll a s e n s e f o r t h e t r i g o n o m e t r i c s y s t e m [ 2 ]. A t t h e s a m e t i m e , ( 2 ) c a n b e e s s e n t i a l l y s t r e n g t h e n e d f o rl a c u n a r y s y s t e m s , f o r e x a m p l e , t i m R a d e m a c h e r s y s t e m

r k ( t ) = s g n s i n 2 k - 1 7 r t ( k --- 1 , 2 , . . . ) .R e c a l l t h a t a B a n a c h s p a c e X o f m e a s u r a b l e f i m c t i o n s x = x ( t ) o n [0, 1] is r e a r r a n g e m e n t - i n v a r i a n ti f t h e r e l a t io n s y * ( t ) < _ x * ( t ) a n d x E X i m p l y t h a t y E X a n d [lY[Ix _ 0 i s a c o n t i n u o u s c o n v e x f i m c t i o n o n [0 , r s u c h t h a t S ( 0 ) = 0 t l m n t h e O r l i c z s p a c e L sc o l m i s t s o f a l l x = x ( t ) s u c h t h a t

[ IX [[Ls = in f u > 0 : S d t < l < o c .0

Samara . T rmm lated from b: i b i r s k~ Mat em at i ches k i ~ Zhur na l , V ol. 41, No. 4 , pp. 72 9-739, July-A ugm st, 2000.Original ~wticlesubmit ted December 22, 19.98.0037-4466 /00/4104-0601 $25 .00 (~ 2000 Kluwer Academ ic /P lenum Pub l i shers 601


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I f ~ ( t ) > 0 i s a n i nc r e a s i ng conc a ve func t i on on [0 , 1] t he n t he M a r c i nk i e w i c z spa c e M ( ~ ) c ons i s t so f a l l x = x ( t ) s uc h t h a t1 ,I I x l I M ( , ) = sup x ( s ) d s : 0 < t < 1 < o c ,

oan d th e Lorentz space Ap (~) (1 < p < r coI r si s ts of a l l x = x ( t ) f o r w h i c h

I I x l t A , , ( ) = z * ( t ) ] " d ( t < c r0

I n p a r t ic u l a r, w e d e n o t e A I ( ~ ) b y A ( ~ ) .S upp os e t ha t N ( u ) = e t ? - 1 , L N i s the O r l i c z s pa c e c ons t r uc t e d f o r t h i s f l nm t i on , a nd G i s t hec losure of Loo in LN . V . A . R o d i n a nd E . M . S e m ~ n ov [4] de m oi r s t r a t e d t h a t t he s y s t e m {rk}k~176 na t t l S X is e qu i va l e n t t o t h e s t a n da r d ba s is i n 12 i f a nd on l y if X D G . E qu i v ' a l enc e o f t he s y s t e mOl 3{ r k} k= 1 i n X t o t he s t a nd a r d ba s i s i n 12 i m p l i e s e x i s t e nc e o f a c ons t a n t C > 0 s uc h t h a t

C - 1 1 1 ( a k ) l l 2 < _ _ akrk C I l ( a k ) l l 2 ,I k = l X

( 3 )

,,,here II(a~,)ll2 = ( E ~ = , a 2 )1 / 2- S . M ont gom e r y - S m i t h [ 51 l a t e r s t r e ng t he ne d t h i s r e s u l t by f i nd i ngt h e d i s t r i b u t io n o f th e R a ~ t em a c h e r s tu n s . T o t h i s e n d , h e u s ed t h e J ~ - f m m t i o n a l f or th e B a n a c hpa i r ( /1 , /2) .M or e ove r , if t he no r m o f t he R I S X i s o r de r s e m i c on t i nuous ( i. e. , t h e r e l a t i on s x n = x n ( t ) > 0a n d x n ( t ) T x ( t ) a l m os t e v e r yw he r e on [ 0, 1] a s n - - , oo i m p l y t ha t I ] x n H x - - , ] ] x l l x ) t h e n t h e s u b s p a c ege n e r a t e d I ~v t he R a d e m a c he r s y s t e m i s c om pl e m e n t e d i n X i f a nd o n l y i f G C X C G ~ ( s ee [6] o r [7 ,2 . b .4 ] ) . H e r e a nd i n t he s e que l , X ~ is t he dua l o f a R I S X a nd c ons i s t s o f a l l y = y ( t ) s u c h t h a t

I l Y l l x ' = s u p { f f o Y ( t ) x ( t ) d t : l l x H x ~ 1 } < r 1 6 2

F u r t h e r m o r e , n o t e t h a t i f G C X c G ' t h e n t h e o r t h o g o n a l p r o j e c t i o nOO

P x ( t ) = E C k ( X ) r k ( t )k = l

i s b o u n d e d i n X ; h e r e C k (X ) = f ~ x ( t ) r k ( t ) d t a r e t i l e F ou r i e r - R a de m a c l m r c oe f f i ci e n t s o f a f m l c t i on x .T h e a b o v e r e s u l t s i m p l y t h a t t h e o p e r a t o r F x - ( C k (X ) ) ~ .= i a c t s b o u n d e d l y f ro m G ~ i n t o 12.I n d e e d , o w i n g t o ( 3 ) a n d b o u n d e d n e s s o f P i n G r, w e h a v etlFx ll2 = I](c~.(x))~=,ll2 _< C H P x I I G , < C I I P I l G , _ . c , , H x I I G , .

B y t i le de f i n i t ion o f ~ - f un c t i o na l . i n t he e a s e o f t he R a ( l e m a c he r s y s t e m i lL s te ad o f (1 ) w e t ir o sob t a i n t he f o l l ow i ng i ne qua l i t y :~ / ( t , F x ; c o, 12) < A 2 . ) E ( t , x ; L , , G r ) ( t > 0 ) . ( 4 )

L e t u s r e p l a c e t he . Z ( - f unc t i ona l s i n t h i s r e l a t ion w i t h e qu i va l e n t e xp r e s s i ons .6 0 2


3/10

T h e O r l ic z s p a c e L N i s w e l l k u o w n [8] t o c o i n c i d e w i t h t h e M a r c i n k i e w i c z s p a c e M ( ' g ' l) w i t h'r (u) = u lo g 1/2 e / u . T h e r e f o r e , G ' = A ( 'r [ 3, p . 1 5 8] . S i u c e L 1 i s a l s o t h e L o r e u t z s p a c e c o n s t r u c t e df o r t h e f u n c t i o n Co(u) = u , we h a v e

1L 1 , G ' ) ~ / x * ( s ) d ( , n i u ( s , tslog 1/2) E ( t , e / 8 ) );

. 10[3 , p . 1 6 4 ] ( t h i s m e a z ls v a l i d i t y o f i n e q u a l i t ie s l i k e ( 3 ) w i t h s o m e c o n s t a n t i n d e p e n d e n t o f x E L 1 a n dt > 0 ) . A f t e r s i m p l e t r a n s f o rm a t io n s w e co n c l u d e t h a t

v(t ) 1x / x ' ( s ) d s + t / x * (s )l o g 1/2 e / s d s , (5 )g ( t , x ; L1 , G )

,J

o ~,(t)w h e r e u ( t ) = e x p ( 1 - 1 / t 2 ) .

P u t t i n g t = 1 / V ~ , (n = 1 , 2 , . . . ) , f r o m ( 4 ) , ( 5 ), a u d t h e r e l a t i o nx ~ (,~);c0,1~ ~

k---1r e s u l t i n g f r o m H o h n s t e d t ' s f o r m u l a [ 9], w e a r r i v e a t " th e i n e q u a l i t y{ J J }~ - '~ I 1 x , ( s ) lo g l / 2 e / .s d[ 4 ( ~ . ) ] ~ ~ ' / ' ~ ' - "< A 2 x * ( s ) d s + - ~~ / ~ k k = l 0 e l - no r

[c~.(~)]2 A2 v ~ f x*(s)ds + x*(s)logl /2k k = I 0 e l - n

w h e r e t h e c o n s t a n t A 2 > 0 is i n d e p e n d e n t o f x E L1 a n d n = 1 , 2 , . . . .U s i n gt

x**( t) = 7 x* ( s )d s ,0

w e c a a d e r iv e a n i n e q u a l i t y s i m i la r to ( 6) b u t w i t h a s o l e e x p re s s i o n o n t h e r i g h t - h a u d s i d e .F i r s t o f a l l , f o r e v e r y 0 < t < 1 w e h a v e

(6 )

1 t 1 1 1

t 0 t t ut 1= l ~ f x * ( . ) . . + x * ( . ) l ~ 1-eu..

0 tN o t e t h a t f o r 0 < u _< e - 2 w e h a v e l o g 1 /2 e / u < _ l o g 1 / u . T h e r e f o r e ,

1 e - 2 e - 2

/ x * ( u ) l o g l / u d u ~ / x * ( u ) l o g l / u d u ~ _ / x * ( u ) l o g l / 2 e / u d u ;t t t

( T )

6 0 3


4/10

i n o r e o v e r , if 0 < t g e - 3 t h e n1 e - 2J x * ( u ) l o g l / u d u > _ _ x * ( ' u ) l o g l / u d u > _ -

l e-3T hu s, fo r 0 < t 4

2 ( e - 1 ) x * ( e - 2 ) > _ _ 11 f x , ( u ) l o g a / 2e ( e + I ) e-21

1 f x * ( u ) l o g U 2 e / u d u ;>- e (e + 2 )t

e / u d u .

1 e 1 - n 1/ / l / x* ('u )lO gl/2e/'u 'd 'l"x * * ( u ) d u > _ ( n - 1 ) x * ( u ) d u + e ( e + 2 Tel--n 0 el--n

-> e(e+2------~ x/~ x * ( u ) d u + x * ( u )l o g l/ 2 e /u d u .0 e1-n

F i n a l l y , f i ' o m ( 6 ) w e o b t a i n

0 i s i n d e p e n d e n t o f x E L 1 a n d n > 4 .H o w e v e r , it i s p o s s i b l e t o o b t a i n ( a zl d t h i s i s t h e m a i n g o a l o f t h e a r t i c le ) n m c h m o r e e x a c ti n f o rm a t i o n o n t h e F o u r i e r - R a d e m a c h e r c o e f f ic i e nt s o f t h e f u n c t i o n s i n a R I S t h a n ( 6) a n d ( 8 ).G i v e n m e a s u r a b l e h m c t i o n s x = x ( t ) a n d y = y ( t ) o n [0 , 1] , put

d t ( i f t h e i n t e ~ ' a l e . x i s t s ) .1

= /o

o o o QS i m i l a r ly , g i v e n s e q u e n c e s a = ( a k ) k8 8 a n d b = (bk )k=l , p u t( i f t h e s e r i e s c o n v e r g e s ) .

o Q( a , b ) =

k = lI f

T a ( t) = Z a k r k ( t ) {9)k = l

o Qt h e n t i l e f o l lo w i n g r e l a t i o n h o l d s f or e v e r y f l m c t i o n x = x ( t ) a n d e v e r y s e q u e n c e a = ( a k )k = 1 w i t hf i n i t e l y m a n y n o n z e r o t e r m s :

1(F x , a ) = Z C k ( X ) a k = / x ( t) Z a k r k { t ) d t = ( x , T a ) . (10 )

k = l 0 k = l

60 4


5/10

B y d u a l i t y , w e e a s i ly o b t a i n a d e s c r i p t i o n f o r t h e s e t o f t h e F o u r i e r - R a d e m a c h e r co e f fi c ie n t s o fa RIS X i f X C G ' . F i r s t o f a l l, d u e t o t h e K h h i c h i n ' s i n e q u a l i t y [ 10 ] f o r t h e L 1 s p a c e , w e h a v e1v I I T a I I L . , < _ I I T a I I L , < __ I I T a l I L . .

Co nseq uen t ly , i t f ol lows f rom IITaHL2 = {{all2 tha t T i s an in j ec t ive bou mte d op era to r f rom 12 in to L1 .B u t t h e n t h e a d j o i n t T * i s a b o m i d e d o p e r a t o r f r o m L ~ o n t o 12 [ 1 1, p . 2 0]. B y ( 1 0 ) , T * = F a n dh e n c e F ( L o o ) = 1 2 . Since the em be dd ing X D L ~ ho ld s fo r every RIS X on [0, 1] [3 , p . 123] , wec e r t a i n l y h a v e F ( X ) D 1 2 . A t t h e s a m e t i m e , a s s h o w n a b o v e , F ( G ~) C 12 . We thus a r r ive a tP r o p o s i t i o n 1 . I f a r e a r r a n g e m e n t - i n v ~ i a n t s p a c e X i s c on t a i ne d i n G ~ t i t an t h e correspol~ding"s p a c e o f t h e F o u r i e r - R a d e m a c h e r c o e tt ic i en t s i s F ( X ) = 12.W e n o w t u r n t o s t u d y i n g t h e m o r e i n t e r e st i n g c as e o f X D G ~.S i n c e t h e R a d e m a c h e r s y s t e m p o s s es s es t h e S i d o n p r o p e rt y , F is a s u r je c t i v e b o u n d e d o p e r a t o r

from L1 outo co [1, p. 295]; i .e. , F(L ) = c o . ( i i )A t t h e s a m e t i m e , b y t h e a b o v e p r o p o s i t i o n

F ( G ' ) = 1 2 . (12)L e t u s s h o w t h a t t h e s u d e c t i v i t y r e l a t i o n s ( 11 ) a n d ( 1 2) c a n b e " i n t e r p o l a t e d " f o r s p a c e s i n te r -m e d i a t e b e t w e e n G ~ a n d L 1, t i m s p a c e s o f th e F o u r i e r - R a d e m a c h e r c o e f fi c i en t s s t il l q u i t e d e f i n it er e a r r a n g e m e n t - i n v a r i a n t s p a c e s o f s e q u e n c e s .S u p p o s e t h a t ( X 0 , X 1 ) a n d ( Y o, Y 1) a r e B a n a c h p a i r s . A t r i pl e ( X o , X i , X ) o f s p a c e s s u c h t h a tXo f ] X1 C X C X o + X i i s au interpolat ion t ' r iple w i t h r e s p e c t t o a t r i p l e ( Y o , ~ ' I , Y ) , ]Io N Y 1 CY C Yo + 1 /1 , i f every l inear ope ra to r A , ac t in g b ou nd ed ly f i' on i X0 in to Yo an d f i 'on i X1 in to Yt , isa b o u n d e d o p e r a t o r f r o m X i n t o Y . h l o r e o v e r , i f -

I I A I I x - r - < m a = ' c { ll A l [x - - ,Y : i = 0 . 1 }t h e n ( X o , X 1 , X ) i s s a i d t o b e a n e x a c tX i = l ~ ( i = 0 , 1 ) a n d X = Y t h e n w et h e p a i r ( X o , X 1 ) .h i w h a t f o l l o w s , a n i m p o r t a n t r o l edef in i t ion .S u p p o s e t h a t E is a n i d e a l B a n a c hI f ( X o , X i ) i s a n a r b i t r a r y B a n a c h p a i rx E X o + X 1 s u c h t h a t

I l x l l = II

i n t e Tpol a t i on t r i p l e wi th r espec t t o the t r ip l e (Yo , Y1, Y) . I fs a y t h a t X i s a n ( e x a c t ) i n t e r p o l a t i o n s p a c e w i t h r e s p e c t t oi s p l a y e d b y t h e r e a l in t e r p o l a t i o n m e t h o d . W e r e c a l l i tss p a c e ( I B S ) o f t w o - s i d e d n u m e r i c s e q u e n c e s a = ( a j ) j = _ ~ .t h e n t h e s p a c e ( X 0 , X t ) ~ o f t h e J ff . - m e t h o d c o n s i s t s o f a l l( . z ( 2 J , x ; Z o , x ) bl b < 0 r

T h e s p ac e ( X 0, X I ) [ o f t h e J - m e t h o d c o u ta i ns al l x E X 0 + X1 t h a t a d m i t t h e r e p r e s e n t at i o nx - - ~_ "ajj~-c4j ( i n t h e s e n s e o f c o n v e r g e n c e i n X 0 + X l ) ,

w h e r e u j E X o A X l . T h e n o r m i n (X o , X l ) ~ is d e f in e d b yi n f I 1 ( / ( 2 J , X 0 , X l ) ) t ll E ,{ . j }

w h e r e t h e g T e a t es t l o w e r b o u n d i s c a l c u l a t e d o v e r 'a l l s e q u e n c e s { a j } ~ = _ ~ f o r w h i c h ( 1 3 ) h o l d s .605


6/10

I f E i s a n I B S o f t w o- s i de d s e que nc e s a nd a = (OCk)k=_oQ i s a s e q u e u ce o f n o m m g a t i v e n u m b e r s t h e nt he s pa c e E ( ~ k ) cons i s t s of a ll a = (ak)k=_~ s u c h t h a t (akr~k)'~.=_~ e E , a n d IlallE(a~) = ll(akOk)llE.S u p p os e t h a t E D A ( [ '~ ) = / ~ ( m a x ( 1 . 2 - k ) ) ( { 0} # E C Z ( ~ ) = / l ( m i n ( 1 , 2 - k ) ) ) . T h e n t h em a p p i ng ( X 0 , X I ) ~-~ (X o , X I ) ~ ( (X0, X1) ~-* (X0, X1 )E ) d e te rm ines an exax~t in te rpo la t io n func to r .

" ( X o , )h i s m e a ns t ha t , f o r a r b i t r a r y B a na c h pa i r s ( X o , X I ) an d ( l '0 , Y t ) , the t r ip le (X0, X t , " ;ris a n e xa c t i n t e r po l a t i on t r i p l e w i t h r e s pec t t o t he t r i p l e ( Y 0 , } 1 . ( )( ). " " ~'I)E ) (a similar result is ,J 'ali(1f or t h e f i r - m e t h o d ) . T h e c o l l e c t i o n o f a ll s u c h fu n c t o m i s c al le d t h e real w Y/- inteTTvlat ion method(~r method).I n pa r t i c u l a r , f o r 0 < 8 < 1 a n d 1 < p _< oo w e ob t a i n t he c l as s ic a l i n t e r p o l a t i o n s pa c e s

Xl)h,(2_k0d rX o , X ) o,p = ( X o , = ( X o , X , ) 2 _ k 0 ) .2 . T h e M a i n R e s u l t s

A s m e n t i o n e d , t h e s p a c e G ~ c o in c id e s w i t h t h e L o r e n t z s p a c e A(ulog 1/2 e/ u) . There fore , i t i sq-concave for eve ry q > I [12] an d e la s t i c in the sen se of [13] . But then , by C oro l la ry 5 .10 of [13],i n t e r p o la t i o n in t h e B a n a c h p a i r ( G ~, L 1) i s d e s c r ib e d b y t h e ~ / - m e t h o d . T h i s m e a l m t h a t f o r e v e r yh l t e r po l a t i on s pa c e X w i t h r e s p e c t t o t h is pa ir w e ha veX t .~= ( G , L 1 )E ( 1 4 )f o r s o m e IB S E o f tw o - s i d e d s e q u e n c e s w h ic h c a n b e a s s u m e d t o b e a n e x a c t i n t e r p o l a t i o n s p a c e w i t hr e s pe c t t o t he pa i r / *~ = ( l ~ , l oo ( 2 - k ) ) [ 14 ].

T h e o r e m 1 . S u p p o s e t h a t X i s a separable interpola t ion R IS with respec t to the pa ir ( G ' , L t )such that (14) holds. Th en th e corresponding .space F ( X ) of the Fom'ier-Rademacher coet tic ientscoincides with the separable s p a c eR = ( 1 5 )

PROOF. B y du a l i ty [11 , p . 20] , i t suf fice s to ve r i fy th a t th e a~ ljoint ope r a tor F* i s in jec t iv e f i 'om R*i n t o X * .F i r s t o f a ll , w e c a n t a ke t h e B I S E i n ( 14 ) s o a s t o f u l f il l t he c ond it io~ m o f t he dua l i t y t h e o r e m f o rt he r e a l i n t e r po l a t i on m e t hod [ 14 , p . 128 ] :

A ( / ' ~ ) i s e ve r yw he r e de n s e i n E ( 16 )a n d E ~ I r 1 6 2 a nd E ~ / ~ ( 2 - k ) . (17)

To prove (16) , den ote by E o th e c losure of A( I' r162 n E az ld show tha tX ' ~ (18)( G , L 1 ) 0 .

S i nc e X i s s e pa r a b l e , t he s pa c e L ~ i s e ve r yw he r e de n s e i n X [3 , p . 138 ] . T he r e f o r e , by t he de f i n i t i on OOof t he s pa c e s o f t he . X / - m e t hod , f o r e ye D " x E X , t he r e i s a s e que n c e { z n} n = l C L ~ s uc h t ha t

I lxn - x][x = [ l ( , )F(2k,xn - - x ; G ' , L l ) ) k l l E ~ 0as n -* r For each t > 0 t l l e .X( - f imc t iona l J L . ( t , x ; X o . X 1 ) i s a m) t'm on t he space X0 + X1.C o l ~ e q ue n t l y , t he p r e c e d i ng r e l a t i o n i m p l i es t ha t a s n --~ c ~

l l( ' ( 2 k , x , , : e ' , L I ) ) a - ( 9 ( 2 k , x ; V ' , L ) )k l lE " -* O . ( 1 9 )By t im de f in i t ion of th e .d/~. fm wt io na l , we have dC (2 k , Xn; G l, L1) _~ 2kHxnllL1 for k 0 . T h e r e b y f or e a ch n = 1 , 2 , . . . t h e s e q u e n c e ( ~ / ( 2 k , xn; G ~, L l ) )k= _~ l ies60 6


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in A( /*~) and (19) imp l ie s th a t (5~( (2k ,x : ', L 1 ) ) k = - c c 6 E0; i .e . , x 6 ( G ' , ~1 ) E o . R e l a t i on ( 18 ) i sp r ove n : he nc e, w e m a y a s s um e t h a t ( 16 ) i s s a ti s f ie d .W e now t u r n t o ( 17 ) . S i nc e I lYIIL~


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T h e o r e m 2 . S u p p o s e f l i n t R i s a set )ara lf le Ban ach space o f sequences wh ich i s an in terpo la t ionspace , wi th respec t to the pa ir ( /2, co) and sat is t ies ( 1 5 ) . T h e n t im e q u a l i ty F ( X ) = R h o l ds f or e v e r yseparab le R I S X s a t i s ( v ' i n g ( 1 4 ).T h e l a s t a s s e r t i o n s h a r p l y c o n t r a s t s w i t h t h e c a s e o f X C G ' c o n s i d e r e d i n P r o p o s i t i o n 1 .

. T h e o r e m 3 . S u p p o s e t h a t Xo m i d X1 a r e s e p a x ab l e i n t e rp o l a t io n R I S ' s o n [ 0, 1 ] w i th r e s p e c t t ot h e p a i r ( G ' , L I ) . I f the correspo ud ing spaces F ( Xo ) and F ( X1) o f t h e F o u r i e r - Ra d e m a c h e r c o e f f ic i e n tsco inc ide then X 0 = Xl and the ir norn~ are equ iva len t .

P R O O F . A s w a s o b s e r v e d b e f or e T h e o r e m 1 , i n t e r p o l a t i o n i n t h e p a i r ( G ' , L 1 ) i s d e s c r i b e d b yt h e 3 ~ / - m e t h od . T h e r e f o r e , t h e r e e x is t B I S ' s E 0 a n d E 1 o f t w o - s i d e d s e q u en c e s w h i c h a r e e x a c ti n t e r p o l a t i o n s p a c e s w i t h r e s p e c t t o t h e p a i r l*cr a n d s u c h t h a t

( i=0 , 1 ) .T h e n , b y t h e h y p o t h e s i s a n d T h e o r e m 1,

F ( X o ) = F ( X ) = = ( 12 ,c o ) .

D e n o t e t h i s s p a c e b y R .A s in t h e p r o o f o f T h e o r e m 1 , w e m a y a s s u m e t h a t E i ( i = 0 , 1 ) s a t i s f y ( 1 6 ) a n d ( 1 7 ). C o n s e q u e n t l y ,T ( s e e ( 9 )) is a n i n j e c t i v e b o u n d e d o p e r a t o r f i' om t i f i n t o X ~ a n d X ~ . I n o t h e r w o r d s , f o r e v e r y s e q u e n c ea = (ak)k= 1 6 R l w e h a v eakrk ~ . akrk I la l l~ , .Ik=l X / ) I I k = l [ i X ~

I f ~ i s t h e c lo s u r e o f L ~ i n X ~ ( i = 0, 1 ) then

S i n c e X ~ a n d X ~ a r e ~ Z - i n t e r p o l a t i o n s p a c e s w i t h r e s p e c t t o t h e p a i r ( L N , L ~ ) ( s e e t h e p r o o f o fT h e o r e m 1 ), i n v ie w o f ( 2 1) Y 0 a n d Y1 a r e i n t e r p o l a t i o n s p a c e s w i t h r e s p e c t t o t im p a i r ( G , L ~ ) .C o n s e q u e n t l y , t h e y s a t i s f y t l m c o n d i t i o n s o f T h e o r e m 3 o f [1 8 ]; h e i m e , Y 0 = } ' 1 . F r o m t h e d e f i n i t i o no f t h e d u a l s p a c e w e o b t a i n Y / = X ~ ' ( i = 0 , 1 ) a n d t h u s X ~ ' = X ~ '. I n v i e w o f s e p m ' a b i l it y o f ) t o a n dX 1 , w e h en c e c o l m l u de t h a t X 0 = X 1 a n d t h e i r n o r m s a r e e q u i v a le n t . T h e t h e o r e m i s p r o ve n .3 . C o n c lu d i n g R e m a r k s a n d E x a m p l e s

R E M A R K 1 . S u p p o s e t h a t Z o - { x 6 L 1 : F x = O } m i d Z 1 - - { x 6 G ' : F x = 0 } . T h e n t h eq u o t i e n t s p a c e G ' / Z 1 i s n a t u r a l l y e m b e d d e d i n t h e q u o t i e n t s p a c e L1 /Z o, ~ = x + Z1 ~-~ x + Z o ,a n d t h i s e m b e d d i n g i s c o n t i n u o u s w i t h re s p e c t t o t h e q u o t i e n t n o r m . D e f i n e t h e l in e a r o p e r a t o r

O 0o n L 1 / Z o a s fo l lows: F'~ . = F (x + Zo) = F x - ( ck (x ) )k= 1 . I u v i e w o f ( 1 1 ) a n d ( 1 2 ) , t h i s o p e r a t o r i sa n i s o m o r p h i s m b e t w e e n t h e s p a c e s L 1 / Z o a n d c o , a s w e l l a s b e t w e e n G ' / Z 1 a n d 1 2 . C o n s e q u e n t l y ,.)g~ t, 5:: G ' I Z , , L 1 / Z o ) ~ ( t , F~ ; 12, c o) , (22)

w i t h s o m e c o n s t a n t i u d e p e n d e n t o f/ ~ e L 1 / Z o a n d t > 0 .S u p p o s e t h a t a B I S E o f t w o - s i d e d s e q u e n c e s s a t i s f i e s ( 16 ) a n d ( 1 7 ) . T h e u , t \ v T h e o r e m 1 , t h eI . iFo p e r a t o r F i s a n i s o m o r p h i s m b e t w e e u t il e s p a c e s ( G , L 1 )E / Z E an d ( /2 , C0)E ' , w her e Z E = { x 6

( c ' , : F x = 0 } . T h e , ' e f o r e , b y" ( 2 2 )( G ' / Z 1 , L , / Z o ) ~ " = ( G ' , L 1 ) ~

608


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( t h e n o r n r s a r e e q u i v a l e n t ) .R E M A R K 2 . B y K h i n c h i n ' s i n e q u a l i t y [ 1, p p . 1 5 3 - 1 5 4 ] , t i m e q u a l i t y F ( L p ) - 12 i s v a l i d f o r

1 < p < o c . N e v e r t h e l e s s , t h e a s s e r t i o n s o f T h e o r e m s 1 a n d 2 b e c o m e f a l s e i f w e r e p l a c e G p i nt h e i r s t a t e m e n t s w i t h L p f o r s o m e p i n t lL is i n t e r v a l . I n d e e d , ( L p , L 1 ) o ,q = L q i f 0 E ( 0 , 1 ) i s suc ht h a t q = p / ( p O - 0 + 1 ) [ 21 , p . 1 57 ]. A t t h e s a m e t i m e , t l m e q u a l i t y ( /2 , co)o,q = Ir ,q h o l d s f o r t h es a m e 0 a n d q , w h e r e r - - - - 2 q ( p - 1 ) / p ( q - 1 ) [2 1 , p . 1 4 6 ] . S i n c e q < p , w e h a v e r > 2 a n d t h e r e f o r eF ( L q ) = 12 ~ Ir ,q .T h e a b o v e re s u l ts e n a b l e u s to f in d r e a r r a n g e m e n t - i n v a r i a n t s p a c e s o f f u n c t i o n s w i t h g i v e n s p a c e so f t h e F o u r i e r - R a d e m a c h e r c o e f fi ci e nt s. W e e x lf ib i t t w o e x a m p l e s .

EXAMPLE 1 . S u p p o s e t h a t R = I p, 2 < p < c r S i n c e Ip = ( /2 , c0)0,p , w h er e 0 = ( p - 2 ) / t ) [21,p . 146 ] , t he spa c e X p w i t h F ( X p ) = Ip i s d e t e r m i n e d b y t h e f o r m u l a

X p = ( a ' , L 1 ) 0 , p .T h e s p a c e s G ' a n d L 1 c o i n c i d e w i t h t h e r e s p e c t i v e L o r e n t z s p a c e s A ( u l o g 1/2 e / u ) a n d A ( u ) . T h e r e f o r e ,

L f = ( L 1 , L ~ ) 4 2 _ k ) ;' = ( L 1 , C ~ ) l l( 2 _ k ( l _ k ) - l / 2 ) , L 1m o r e o v e r , s i nc e w e c o n s i d e r R I S o f f m m t i o u s d e f i n e d o n [0 , 1 ], a s t h e p a r a m e t e r s o f t h e J - m e t h o dw e t a k e B I S 's o f s e q u e n c e s a = ( a j ) j - - ~ [2 2]. A p p l y i n g t h e r e i t e r a t i o n t h e o r e m f o r t h e f f ~ - m e t h o d , w en o w i n fe r t h a t X p c o i n c i d e s w i t h t h e L o r e n t z s p a c e A p ( ~ p ) , w h e r e ~ p ( u ) = u l o g W p c / u .

E X A M P L E 2 . S u p p o s e t h a t R = A ( 1 / k ) i s t h e L o r e n t z s p a ce o f a l l s e q u e n c e s a = ( a k ) k = l f o rw h i c h t l m n o r m OQ ,I l a l l , ( i / k ) = k = Ii s f i n it e . T h i s s p a c e i s s e p a r a b l e . C h o o s i n g a s u i t a b l e B I S E o f s e q u e n c e s , w e c a n c l e m o n s t r a t e t h a t( /2 , c 0 ) ~ = A ( 1 / k ) . H o w e v e r , w e p r o c e e d a l t e r n a t i v e l y , u s i n g E x a m p l e 2 o f [ 1 8] ( s e e [1 9 ] f o r d e t a i l s ) .O b s e r v e t h a t A ( 1 / k ) * - - A ( 1 / k ) ' - - I x (l o g ) a n d l l ( l o g ) t = - A ( 1 / k ) , ( 2 3 )

o ow h e r e t h e M a r c i n k i e w i c z s p a c e I I ( lo g ) c o n s i s t s o f a ll s e q u e n c e s a = ( a k ) k = 1 f o r w h i c h t h e n o r mk

I la l l , ( o ) = s u p l o g :2 - 1 ( 2 ) a ;k = l , ' , . . .. , = 1

i s f i n i t e [ 2 3 , p . 1 90 ]. S i n c e / 1 (l o g) i s a n i n t e r p o l a t i o n s p a c e w i t h r e s p e c t t o t h e p a i r ( l l , / 2 ) ( s e eE x a m p l e 2 o f [1 8] ), A ( 1 / k ) i s m i i n t e r p o l a t i o n s p a c e w i t h r e s p e c t t o t h e p a i r ( /2 , c o ) b y ( 2 3 ) . T h u s , b yT h e o r e m 2 . F i s a s u r j e c ti v e o p e r a t o r f r o m s o m e s e p a r a b l e R I S X t o A ( 1 / k ) . M o r e o v e r , i ts a d j o i n t Ti s a n i n j e ct io n f r o m A ( 1 / k ) ' = / l ( l o g ) t o X ' . F r o m t h e r e s u l ts o f [ 18 ] w e t h e n o b t a i n X ' = M ( ~ ) ,w h e r e ~ ( u ) = u l o g 2 1 o g 2 ( 1 6 / u ) . I n v i e w o f s e p a r a b i l i t y o f X , w e h a v e X = A ( ~ ) [ 3, p . 13 8 ]. T h u s ,F ( A ( ~ ) ) = A ( 1 / k ) .

R e f e r e n c e s1 . K~w zmarz S . and S te inha us f i . , Th(u)rv of Orthogoual Series [R ussian translation] , Fizmatgiz, Mix, ow (1958).2. Ovchinaikov V. L , Rt~spopo~n V. D. , and R od in V. A. , "Ex act estimates for tim F ourier coefficients of integrab lefunctions and 5g. f imctionals," Mat. Zam etki, 32, N o. 3, 292-302 (1982).3 . K re~n S . G . , Pe tu n in Y u . L , an d Sem~ nov E . M. , Inte rpo lation of Linear Ope rators [ in RtL~sian], Nauka, Mo scow(1978).4 . Rodin V. A. and 8emyo nov E. M. , "Ra dem ~h er ser ies in sym me tr ic spaces ," A na l Math . , 1 , No. 3 , 207-222 (1975) .

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5. Montgomery-Smi th S . , "Th e d is t r ibu t ion o f Rad ema eher sums ," Proc . Amer . M ath . Soc ., 109 , No . 2 , 517-522(1990).6 . R o d i n V . A . a n d S e m 6 n o v E . h L , "Com plementab i l i ty o f the subspace genera ted by t im R,~le ,nacher s ys te m ina r ear r angement - invar ian t spew.e ," Fnnk ts ional . Anal . i P r i l~ he n . , 13 , No . 2 , 91-92 (1979).7. L inde ns t rav~ . I . and Tzaf r i r i L . , Classical B anach Space s. Vo| . 2 : F unction Spaces, Spr inger-Ver lag, Ber l in (1979) .8. Rut i tski[ Ya. B. , "On so me c lasses o f nmasurab le fmm tions ," Uspekh i M at . N auk , 20 , No . 4 , 205-208 (1965).9. Hohns tedt T . , "Inte rpol at io n of qm~si-normed spaces," M ath. Scand ., 26, No. 1 , 177-199 (1970) .

10 . Szs rek S . J . , "On the be s t co ns tan ts in the K hinch in inequal i ty ," S tud ia Math . , 58 , 197-208 (1976).11 . Pietsch A. , Operator Ideals [Russian translat ion] , Mir , Moscow (1982) .12 . Novikov S . Ya . , "Co type an d typ e o f Loren tz funct ion spaces ," Ma t . Zam etk i , 32 , No . 3 , 213-221 (1982) .13 . Kal ton N. J . , ~Calderon coup les o f r ear r angement inva r ian t spaces , ~ S tud ia M ath . , 103 , No . 3, 233-277 (1993).14 . BrudnyT Yu. A. a n d Kr u g l y a k N . Y a . , "Functo r s o f r ea l in terpo la t ion ," subm i t ted to VINITI , 1981, No . 2620-81 .15 . Ovchinnikov V . L , "Th e m eth od o f o rb i t s in in terpo la t ion the ory ," M ath . R ep . S er . 1 , No . 2, 349--515 (1984). -16 . Kr~.snosel'skiT M. A . an d R ut i tsk f f Ya. B. , Convex Fnnctions and Orlicz Spaces [ in Rt~sian] , Fiz,natgiz, Moscow(1958).17 . Kantorov ich L . V . and Ak i lov G . P . , Funct ional Ana lys i s [ in Rt~s ian] , Nauka, M oscow (1977) .18 . As tashk in S . V . , "On ser ies w i th r espect to the Rademacher sys tem in r ear r angement - in~r ian t spaces ' c lose ' toL~r Fnnktsional . An al . i Pr i lozhe n., 32, No. 3 , 62-65 (1998) .19 . Asmshk in S . V . , "On in terpo la t ion o f subspaces o f rearr angement- in~}~r ian t spac es g enera ted ~ , Rade ma cher sys -tem ," Izv. Ross. Akad . Estes tv . N auk Ma t. Ma t. Mo del . Inform. Upravlen. , 1 , No. 1 , 18-35 (1997) .20. Sparr G ., "Interp olat ion of weighted Lp-spaces, ~ Stud ia M ath. , 62, 229-271 (1978).21 . Tdebe! H . , Interpo lat ion The ory : Fun ction Spaces; Dif ferentia l Op erators [Ru ssian translat ion] , Mir , Moscow (1980) .22 . Dmi t r i ev V. L, Krei'n S. G . , and Ovchinnikov V . I . , " F n n d m n en t a l s o f th e i n t e r p o l a t io n t h eo r y o f li n ear o p e r a t o r s , "in: M ezhvnz. Tem at. Sb. Na uch . T rudov [ in Rt~sian] , Yaroslavsk. Univ. , Yaroslavl ' , 1977, pp. 31-74.23 . Gokhberg I . Ts . and Kre}'n h L G., An In t roduct ion to the T heory o f L i,mar Nonsel f ad jo in t Opera to r s in t t i lbe r tSpace [in Russian], Nank~, l~,Iosc.ow 1965).

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