S. V. Astashkin- On Cones of Step Functions in Rearrangement Invariant Spaces

8/3/2019 S. V. Astashkin- On Cones of Step Functions in Rearrangement Invariant Spaces

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O N C O N E S O F S T E P F U N C T I O N S I NR 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 1 7 . 9 8 2 . 2 71 . W e r e c a l l t h a t a 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 ( r. i. s p a c e ) is a B a n a c h s p a c e X o f m e a s u r a b l ef u n c t i o n s o n [0 , 1] f o r w h i c h

( 1 ) f r o m y e X a n d I x ( t) [ < [ y ( t ) f i t f o l lo w s t h a t x e X a n d l [ x t l x < I [y l [x ;( 2 ) i f y e x a n d ] x ( t) ] i s e q u i m e a s u r a b l e w i t h [ y ( t ) l , t h e n x e X a n d ! ] x ! ] x = I l y N x .

T h e t h e o r y o f r .i . s p a c e s i s p r e s e n t e d i n [1 ].L e t X b e a n r . i . s p a c e o n [0 , 1 ]. T h e n i t i s u n i q u e l y d e t e r m i n e d b y t h e c o n e o f f u n c t i o n s y E X

o f th e f o r m (DOy ( t ) = Z c ~ x ( 2 - ~ , 2 - ~ + l l ( t ) ,

k----1w h e r e ck >_ O, Ck 0 )

O < s < m i n ( 1 , 1 / t ) ~(S)S a m a r a . T r a n s l a t e d f r o m S i b i rs k i ~ M a t e m a t i c h e s k i ~ Z h u r n a i , V o l . 3 4 , N o . 4 , p p . 7 - 1 6 , J u l y - A u g u s t , 1 99 3.O r i g i n a l a r t i c l e s u b m i t t e d A p r i l 2 , 1 9 9 2 .

0 0 3 7 - 4 4 6 6 / 9 3 / 3 4 0 4 - 0 5 9 7 $ 1 2 .5 0 ( ~) 1 99 3 P l e n u m P u b l i s h i n g C o r p o r a t i o n 5 9


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d e t e r m i n e s t h e d i l a t i o n f u n c t i o n f o r ~ ( t ) . T h e b e h a v i o r o f .M ~ , (t ) c h a r a c t e r i z e s t h e l o w e r a n d u p p e rd i l a t i o n e x p o n e n t s I n ~ 4 ~ ( t ) I n A 4 ~ ( t )7 ,p = l i m 5 ,p = l i m (2 )t - , 0 + l n t ' t- ~o o l n tI n p a r t i c u l a r , i f w e t a k e a s ~ ( t ) t h e f u n d a m e n t a l f u n c t i o n ~x( t ) = ] l x (o , t ) l I x o f a n r .i . s p a c e X ( f o rL , , i t e q u a l s t l /P ; f o r A ( q ~ ) , q a ( t ) ; a n d f o r M ( ~ ) , ~ ' ( t ) = t / ~ ( t ) ) , t h e n w e o b t a i n t h e n u m b e r s 7 x = % , xa n d 5 x = 5 ~,x " I f w e s u b s t i t u t e t h e n o r m [ l o t l l x - . x , o f t h e d i l a t i o n o p e r a t o r , f o r .M ~ , ( t) i n ( 2 ) , t h e nw e a r ri v e a t t h e d e f i n i t io n o f t h e B o y d i n di c e s a x a n d / 3 x o f t h e s p a c e X .

e (x)2 . I n w h a t f o l l o w s w e c o n s i d e r s y s t e m s { k}k=l o f s u b s e t s o f [ 0, 1 ] w h o s e m e a s u r e s # ( e k ) = a ks a t i s f y t h e r e l a t i o n sC(3

0 < ak+ l _ 0 ( k = 1 , 2 , . . . ) ; i . e ., t h e r e e x i s t s a n A > 0 i n d e p e n d e n t o f { C k } a n d s u c ht h a t I

k = l X I X= l X k = lP R O O F . G i v e n a s y s t e m o f p a i r w i s e d i s j o i n t s e ts A = { gk }k ~= l, g k C [ 0 , 1 ], p u t T zx t o d e n o t e t h e

a v e r a g i n g o p e r a t o rOO 1 / x ( s ) d s X g , ( t) .2a x ( t ) . ( g , ) Ji= 1 gi

S inc e H T A H L o o _ o L o o ~ -- ] I T A [ IL I - . .. ~ L I ~- 1 a n d si n ce A ( r a n d M ( r a r e e x a c t i n t e r p o l a t i o n s p a ce s b e t w e e nL 1 a n d L ~ [1 , w 5 ] , w e h a v e

I I Y l l A ( ) A ( ) 1 , I IT ~ l lM ( ~ )~ M ( ~ ) 1 .T h e r e f o r e , T zx i s b o u n d e d i n X .F r o m t h e c o n d i t i o n s o f t h e t h e o r e m i t f ol lo w s t h a t t h e v a l u e E T = l ckxek X d e p e n d s o n l y o n t h en u m b e r s c k _> 0 a n d # ( e k ) [4 ]. T h e r e f o r e , w i t h o u t l o s s o f g e n e r a l i t y , w e c a n a s s u m e t h a t t h e s e t s e k( a n d , s i m i l a r l y , f k ) a r e p a i r w i s e d i s j o i n t .598


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O n t h e i m a g e o f T A I ( M ( ~ ) ) , w e d e f i n e t h e o p e r a t o r S :

A s w a s s how n in [4 ], i f 0 < 7 r -< 5 r < 1 the n

4 a (r kfo r a l l d k ( k = 1 , 2 , . . . ) . T h e r e f o re , t h e o p e r a t o r S T & i s c o n t i n u o u s o v e r X a n d

w h e r e A 1 > 0 i s i n d e p e n d e n t o f c k , k = 1 , 2 , . . . . B y i n t e r c h a n g i n g A 1 a n d A 2 , w e s i m i l a r l y o b t a i n

f o r s o m e A 2 > 0 .D e n o t e b y I ( r t h e c la s s o f a l l r . i. s p a ce s t h a t a r e i n t e r p o l a t i o n s p a c e s b e t w e e n A ( r a n d M ( , ~ ).T h e o r e m 2 . L e t a s y s t e m {ek}~ '=l , ek C [0 , 1] , s a t i s f y c o n d i t i o n s (3 ) a n d ( 4 ), X 0 E I ( r a n d

0 < 7r


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I I ( a j ) l l F ( v ~ ) = l l ( a ~ v j ) l l F ) T h e / ( : - m e t h o d s p a c e ( X o , X 1 ) ~ ( w h i c h i s a n i n t e r p o l a t i o n s p a c e b e t w e e nX 0 a n d X 1 ) c o n s i s t s o f x C X 0 + X 1 w i t h f i n it e n o r m

I l x l l = ] l ( E ( 2 k , z ; X o , X l ) ) l l E ,w h e r e

Y . ( t , x ; X o , X l ) = in f { l lx o l lx o + t ! l x i l l x l } .T,=XO+Xl~z i 6 X i

H e n c e f o r t h i t w i ll b e c o n v e n i e n t , f o r t h e t i m e b e i n g , to d e a l w i t h 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 so f f u n c t i o n s d e f i n e d o n t h e h M f - a x e s ( 0, c ~ ) . F r o m t h e e q u a l i t yt

] C ( t , x ; L , , L o o ) = f0

( t > 0 )

i t f o l lows tha t , f i < 0 < 1 , 1 < p < cx~, an d 10 ,p = l p ( 2 k ( e - 1 ) ) ,A (t 0) (L1 lc= , L e e ) t o , a ,

M ( t l - e ) ( L 1 , lc= L ~ r( 6 )( 7 )

w h e r e t h e L o r e n t z a n d M a r c i n k i e w i c z s p a c e s o n ( 0 , cx~ ) are d e f i n e d b y a n a l o g y w i t h t h o s e o n [0 , 1 ] [ 1]F i r s t , w e p r o v e a n a u x i l i a r y c l a i m o f i n d e p e n d e n t i n t e r e s t .P r o p o s i t i o n . L e t X b e a n r . i. s p a c e o n (0 , c r X = ( L 1 , L ~ ) ~ , a n d t e t l ~ r E . T h e n

1 1 ~ 2 ~ l l x ~ 2 k l l P - k l l E ( k = 0 , 4 -1 + 2 , . . . ) ,w h e r e P k ( a j ) = ( a j+ k )j ~ 1 7 6 i s t h e s h K t o p e r a t o r o n E .

P R O O F . W e w i ll s h o w t h a t1 1 ~ 2 ~ x l l x = 2 k l l P - k ( f ( 2 J ) ) l l Ef o r x E X , w h e r e f ( t ) = E ( t , x ; L 1 , L ~ ) . I n d e e d , w e h a v e ( s )

1 1 ~ 2 ~ x l l x = I I (1 C ( 2 J , x ( 2 k s ) ; L 1 , L ~ ) ) j l I E = x * ( 2 - k s ) d s = 2 k l l P _ k ( f ( 2 J ) )l lE .

I n p a r t i c u l a r , e q u a l i t y ( 8 ) y i e l d sl [~2ktl ~ 2k l lP- kl] .

F u r t h e r m o r e , s i n c e E i s i n t e r p o l a t o r y b e t w e e n l oo a n d / o ~ ( 2 - k ) , t h e r e e x i s t s a C1 > 0 s u c h t h a t , f o re a c h a = ( a j ) e E , t h e s e q u e n c e ~ = ( a k ) ,~k = K:(2k, a ; l o o , l o o ( 2 - J ) ) = s u p [ m i n ( 1 , 2 k - J ) [ a j l ] ,j=O,+l, . . .

b e l o n g s t o E t o o a n d I I ~ I I E ~ < C I l I a l I E [ 7 , w 7 ] .O b s e r v e a l s o t h a t ] a k t ~ _ - d a n d t h a tl i m ~ k = 0

k - - - - + - - ~( 9 )

600


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b y th e m o n o t o n i c i t y o f { ~ k} a n d t h e c o n d i t io n loo ~ E . F o r a ~ E , w e i n t r o d u c e t h e f u n c t i o nfa(t) = ~F_,k~176 ~X[2* ,2*+~) ( t ) . S i nce


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T h u s ,c k o , I I ( c k ) l l t l .

k = l XD e n o t e b y Y t h e s p a c e o f a l l m e a s u r a b l e f u n c t i o n s x = x ( t ) o n [ 0, 1] w h o s e e x t e n s io n s b y z e r o o n to[0 , cx~) b e l o n g t o X , t h e s p a c e Y b e i n g e n d o w e d w i t h t h e n o r m e q u a l t o t h e n o r m o f t h e e x t e n s i o n i n

X . T he n Y i s an r . i . spac e on [0 , 1 ], 7Y = a y = f l y = 5 y = 1 /2 , a n d ( 1 0 ) h o ld s . T h e s p a c e s 1 1 (2 k / 2 )a n d E = G ( 2 - k / 2 ) a r e i n t e r p o l a t i o n s p a ce s b e t w e e n lo~ a n d / c r - k / 2 ) [ 9 ] . T h e r e f o r e , i f w e s u p p o s eth a t Y = A ( t 1 /2 ) o n [ 0, 1 ], t h e n , a r g u in g a s i n t h e s e c o n d p a r t o f t h e p r o o f o f t h e P r o p o s i t i o n , w ec a n sh o w th a t / 1( 2 - k /2 ) z _ = E z _ , i . e . , G z _ = l l [ z _ ( s e e a l s o [ 1 0 , P r o p o s i t i o n 2 ] ). T h e l a s t f a c tc o n t r a d i c t s t h e c h o i c e o f t h e s p a c e G . T h e t h e o r e m i s p r o v e n .

4 . D e n o t e b y F ( r t h e c l as s o f t h o s e r .i . s p a c e s o n [0 ,1 ] w h i c h h a v e f u n d a m e n t a l f u n c t i o n rW e w i l l s h o w t h a t t h e c l a i m o f T h e o r e m 2 f a il s f o r F ( r e v e n i f w e re p l a c e t h e e q u i v a l e n c e o f n o r m so n t h e c o n e o f f u n c t i o n s o f th e f o r m ( 1) b y t h e i r i s o m e t r i c c o in c i d e n c e :W e r e c a ll t h a t , f o r t h e d i l a t io n e x p o n e n t s o f t h e f u n d a m e n t a l f u n c t i o n a n d t h e B o y d i n d i c es oa n r . i. s p a c e , w e a lw a y s h a v e 0 < a x _< 7 x -< 5 x _ < /3 x < 1 . F o r t h e m o s t i m p o r t a n t c l a ss e s o f r . is p a c e s ( L o r e n t z s p a c e s , M a r c in k i e w ic z s p a c e s, a n d O r l i c z s p a c e s ) , w e m o r e o v e r h a v e a x = 7 x a n d~ x = 5 x . N e v e r th e l e s s , i n [ 1 1 ] w a s p r e s e n t e d a f i rs t e x a m p le o f a n r . i. s p a c e o n [0 , 1] f o r w h ic h t h el a s t f ai ls . ( S i n c e t h e n s u c h r. i. s p a c es h a v e b e e n c a l l e d t h e s p a c e s o f n o n f u n d a m e n t a l t y p e . )

H e r e w e c o n s t r u c t a n r . i . s p a c e X o f n o n f u n d a m e n t a l t y p e o n [0 , 1] f o r w h i c h ~ x ( t ) = t 1 /2 (he nce7 x = 5 x = 1 /2 ) , a x = 0 , a n d ~ x = 1 , a n d , a t t h e s a m e t i m e , t h e n o r m i n X a g r e es w i t h t h e n o r mi n L 2 o n s o m e c o n e o f s te p f u n c t i o n s o f t h e f o r m ( 1 ) . H e n c e f o r t h , w e d e n o t e

( x , y ) = / x ( t ) y ( t ) d t ,

F o r n C N , n _ > 2 , 1 < i < n , a n d 0 < b _ < l , w e p u t

W n = n - 1 / 2a o = O , a i = 2 2 n ( i - n ) n - 1 , h b = b - l l 2 x ( o , b ) , h i = h a i ,

n

su p h i = n - 11 2 E u i , g n ( s ) = a n w n ( s ) = w . ( ~ l n ) ,i = 1 , . . . , n i = 1

u i = h i x ( a i _ l , a i ) ,r . ( s ) = = w . ( s n ) .

F i r s t o f a ll , w e e s t i m a t e s u p 0 < b < l( g n , hb) a n d s u p 0 < b < l ( r n , hb) ( n = 2 , 3 , . . . ) . F o r 1 < j < n , m a k in gs i m p l e c a l c u l a t i o n s , w e o b t a i n1 j - 1 2 n ( j _ n )f .~ . ui dt = Z ay~ l /2(a i - ai - 1) =0 '


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N o w , i f ai


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a n d , f o r W m = W n m , w e h a v e I l w m l l x = I t w ~ l t 2 = 1 - 2 - 2 n ~ . H e n c e , L 2 C X C M ( t U 2 ) a n d3 / 4 ~ I lw m l lx = I l w m l l 2 ~ 1 ( 1 5 )

f o r a l l m = 1 , 2 , . . . .P r o v e t h a t a x = 0 a n d f i x = 1 . T o th i s e n d , it s u ff ic e s i n v i e w o f ( 1 5) t o c h e c k t h a t H g m H z ,.~ 1

a n d H r ~ l [ x ~ n ~ 1 ( gi n = g n m , r m = r n m ) . I n v i r t u e o f r e l a t i o n s ( 1 2) a n d ( 13 ) a n d t h e d e f i n i t i o n oX , t h e l a s t f a c t i s e q u i v a l e n t t o u n i f o r m b o u n d e d n e s s o f ( g i n , V ) a n d n m ( r m , v ) fo r a l l v e V , I]1)112 < 1a n d m E N .

T h u s , l e t v E V a n d m E N . R e p r e s e n t v a s v = V l + v 2 + v 3 , w i t hm--1 nj nm

1)1 = E E ~ X i , j "~ 5 0 X [ 1 / 2 , 1 ] , " 02 : E c T x i , m ,j = l i =2 i =2V3 : V - - V l - - V 2 .

S i n c e n m a m m _ l


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As a resul t, f rom (16) - (18) we obta in( g i n , v ) _ 4 ( 1 9 )

for all m e N and v e V, Ilvl12 _ < 1. Arguing in the same way, we can show that ( w i n , cr,m v) < 3 form e N and v e V, I l v l 1 2 < 1. Therefore,v ) = 1 ( w i n , 3n m v ) < . ( 2 0 )n~,n

As has a l read y been sa id , re la t ions (19) and (20) imply a x = 0 and f i x = 1 .R e f e r e n c e s

1. S. G. KreYn, Yu. I . Pe tun ia, and E. M. Sem~nov, Interpolat ion o f Line ar Op erato rs [ in Russian] ,Nauka, Moscow (1978).2 . V . M. Malam id, "On cones de termining rear rangement invar iant spaces ," in: Studies in theTh eo ry of Several Real Variables [ in Russian] , Izd at . Yaroslayl . Univ. , Yaroslavl l: 1981 , pp.45-54.3. N. K. Bari , Trigon om etr ic Series [ in Russian] , F izmatgiz, M oscow (1961).4. I . Ya. Novikov, "Sequences of charac ter is t ic funct ions in rearra ng em ent in varian t spaces," Sibirsk.Mat. Zh., 24, No. 2, 193-196 (1983).5. J . Be rgh and J . L5fstr5m , Interp olat ion Spaces. An Intro du ct ion [Russian t ranslat io n] , IVlirMoscow (1980).6. M . A. KrasnosellskiY an d Ya. F . Rutitski~, Convex O rlicz Spaces [in Ru ssian], F izm atg iz, Moscow(1958).7 . V. I . Ovchinnikov, "The orbi t me thod in in terpola t ion theory , " Math . Rep. , 1 ,34 9-51 5 (1984).8 . V. I . Ovchinnikov, "On a connect ion be tween the complex and rea l in terpola t ion methods , "in: Ab stracts : 16 All-Union Conference on Oper ator T heo ry in Fun ct iona l Spaces [ in Russian] ,Nizhni~ Nov gorod, 1991, p. 167.9. S. V. Astas hkin , "D escript ion of the interpolat ion spaces between ( l l ( w ~ an d ( l ~ ( w ~

/ ~ ( w l ) ) , '' M a t . Zametki , 35, No. 4, 794-797 ( t984) .10 . S . V. Astashkin , "A prop er ty of functors of the real in terpola t ion m ethod ," M at . Zam etki , 39 ,No. 3, 393-407 (1985).11. T. Shimo gaki , "A note on norm s of compression o n funct ion spaces," Pro c. Ja pa n Acad . , 46,239-242 (1970).T R A N S L A T E D B Y V . N . D Y A TL O V

S. V. Astashkin- On Cones of Step Functions in Rearrangement Invariant Spaces

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Transcript of S. V. Astashkin- On Cones of Step Functions in Rearrangement Invariant Spaces