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8/3/2019 S. V. Astashkin- On the Multiplier of a Rearrangement Invariant Space with Respect to the Tensor Product
http://slidepdf.com/reader/full/s-v-astashkin-on-the-multiplier-of-a-rearrangement-invariant-space-with 1/3
F u n c t i o n a l A n a l y s i s a n d I t s A p p l i c a t i o n s , V o l . 3 0 , N o . ~ , 1 9 9 6
B R I E F C O M M U N I C A T I O N S
O n t h e M u l t i p l i e r o f a R e a r r a n g e m e n t I n v a r ia n t S p a c e
w i t h R e s p e c t t o t h e T e n s o r P r o d u c t
S . V . A s t a s h k i n U D C 5 1 7 . 9 8 2 . 2 7
L e t x = x ( s ) a n d y = y ( t ) b e m e a s u r a b l e fu n c t i o n s o n I = [ 0, 11 . W e i n t r o d u c e t h e b i h n e a r o p e r a t o r
R e c a l l t h a t a B a n a c h s p a c e E o f m e a s u r a b l e fu n c t i o n s o n I i s s a i d t o b e r e a r r a n g e m e n t i n v a r i a n t
( r . i .) i f f o r a n y y E E t h e r e l a t i o n z * ( t ) < y * ( t ) ( w h e r e z * ( t ) is t h e n o n i n c r e a s in g r e a r r a n g e m e n t o f t h e
f u n c t i o n [ z ( u ) [ [ 1, p . 9 3 ] ) i m p l i e s t h e r e l a t i o n s z E E a n d [ [ zl [ <_ I[Y [[-
I f E i s a n r . i. s p a c e o n I , t h e n t h e c o r r e s p o n d i n g r .i . s p a c e E ( I x I ) o n t h e s q u a r e c o n s i s ts o f al l
m e a s u r a b l e f u n c ti o n s x = z ( s , t ) o n I × I s u c h t h a t z " e E a n d I l X l l E U x l ) = I I x ' I I E .
I f X , Y , a n d Z a r e r .i . s p a c e s o n I , t h e n t h e o p e r a t o r B f r o m X × Y i n t o Z ( I × I ) i s c o n t i n u o u s i f
a n d o n l y i f t h e e m b e d d i n g o f t h e p r o j e c t i v e t e n s o r p r o d u c t X ® Y i n Z i s c o n t i n u o u s [ 2, p . 5 1] . F o r t h is
r e a s o n , B i s c a l le d t h e t e n s o r p r o d u c t o p e r a t o r .
S t u d y i n g t h e o p e r a t o r B i s i m p o r t a n t b e c a u s e it n a t u r a l l y a r i s e s i n s o l v in g a n u m b e r o f p r o b l e m s
r e l a t e d t o g e o m e t r i c p r o p e r t i e s o f r .i . s p a c e s ( e . g ., s e e [3 , p . 1 6 9 - 1 7 1 ; 4 , 5 ] ).
T h e m a j o r i t y o f p a p e r s o n t h e t e n s o r p r o d u c t o p e r a t o r f o r r .i . s pa c e s a re d e v o t e d t o s t u d y i n g c o n t i n u i t y
c o n d i t i o n s fo r t h i s o p e r a t o r i n s p e c i a l c l a s se s o f t h e s e s p a c e s , n a m e l y , i n L o r e n t z , M a r c i n k i e w i c z , O r l ic z
s p a c e s, e tc . [ 2, 6 -9 ] . I n t h i s n o t e w e c o n s i d e r a r b i t r a r y r . i. s p a c e s. L e t u s i n t r o d u c e t h e m a i n n o t i o n o f
m u l t i p li e r o f a n r .i . s p a c e w i t h r e s p e c t t o t h e o p e r a t o r B .
L e t E b e a n r .i . s p a c e o n I . B y th e m u l t i p l i e r o f t h i s s p a c e w e m e a n t h e s e t M ( E ) o f a ll m e a s u r a b l e
f u n c t i o n s z = z ( s ) o n I s u c h t h a t z ® y E E ( I x I ) f o r a n a r b i t r a r y f u n c t i o n y E E . T h e n M ( E ) is a
l i n ea r sp a c e , w h i c h i s a n r . i. s p a c e o n I w i t h r e s p e c t t o t h e n o r m
IIz I IM(E) = sup {f ix ® YIIE (I×~ ) ; I lyI IE <- 1} .
I t i s c l e a r t h a t w e a l w a y s h a v e M ( E ) C E .
C o n s i d e r a n e x a m p l e . L e t E b e t h e L o r e n tz s p a c e A ( ¢ ) , w h e r e ¢ ( u ) i s a n o n n e g a t i v e i n c r e a s i n g
c o n c a v e f u n c t i o n o n ( 0 , 1] . R e c a l l t h a t t h is s p a c e c o n s is t s o f a ll m e a s u r a b l e f u n c t i o n s x = x ( 8 ) s u c h t h a t
I IX l IA ( ,) = * * ( 8 ) d e ( s ) < oo .
A s s u m e t h a t f o r t h e d i l a t io n f u n c t i o n
w e h a v e
. /Y Y ¢ (v ) = s u p { ¢ ( t v ) / ¢ ( t ) , 0 < t < m i n ( 1 , 1 / v ) }
= l i r a ¢ ( t v ) / ¢ ( t )t--.~to
w h e r e t o E [ 0, 11 d o e s n o t d e p e n d o n v E [0 , 1 ]. S i m p l e c a l c u l a t i o n s s h o w t h a t M ( A ( ¢ ) ) = h ( . / / g , ) .
L e t X e ( S ) = 1 , s E e , a n d X e ( s ) = O , s ~ e ( w h e r e e i s a m e a s u r a b l e s u b s e t o f I o r I x I ) .
T h e f u n d a m e n t a l f u n c t i o n C E ( t) = ]lX(o,t) l lE o f t h e r .i . s p a c e E a n d t h e d i l a t i o n o p e r a t o r a t y ( u ) =
y ( u / t ) X [ o , 1 ] ( u / t ) ( t > 0 ) , w h i c h is c o n t in u o u s i n a n y r . i. s p a c e , p l a y a n i m p o r t a n t r o le i n t h e t h e o r y o f
r . i . s p a c e s .
Sam ar a S t a t e U n i ve r s it y . T r ans l a t ed f r om Funk t s i ona l l ny i A na l iz i E go Pr i l ozhen l ya , V o l . 30 , N o . 4 , pp . 58 - 60 , O c t obe r -D ecemb er , 1996 . O r i g ina l a r t i c l e subm i t t ed S ep t emb er 15 , 1994 .
0016- 2663 / 96 / 3004- 0267 $15 .00 C ) 1997 P l enum Pub l i sh i ng C or p or a t i on 267
8/3/2019 S. V. Astashkin- On the Multiplier of a Rearrangement Invariant Space with Respect to the Tensor Product
http://slidepdf.com/reader/full/s-v-astashkin-on-the-multiplier-of-a-rearrangement-invariant-space-with 2/3
T h e o r e m 1 . F o r a n y r . i . s p a c e E o n I w e h a v e
e M ¢ E ) ( t ) = I I ~ , H ~ - . E , 0 < t < 1 ,
I I ~ ' I I M ( E ) - - , M ( E ) = I I G ,I I E - , E , 0 < t < Z , ( 1 )
-- 1
Ha l / , [ IE_ ~E <_ Ha t ] ]M (E)~M (E) _< HatI]E--~E, t > 1. (2)
I n g e n e r a l , i n e q u a l i t i e s ( 2 ) c a n n o t b e r e p l a c e d b y a r e l a t i o n s i m i l a r t o ( 1 ) . T o e s t a b l i s h t h i s i t s u f fi c es
t o c o n s i d e r t h e c a s e i n w h i c h C a ( S ) = s a l n - l ( C / s ) , E = A ( ¢ a ) ( 0 < a < 1 ), a n d C > e x p { 1 / ( 1 - a ) }
a n d a p p l y t h e r e s u l t o f t h e a b o v e e x a m p l e .
R e c a l l t h a t f o r a n y r. i. s p a c e E , t h e l i m i ts
aE = l im ln II~,HE--,E/lnt , ZE = l im l nHa, HE-+E / ln tt--t0 t--~oo
e x i st , w h i c h a r e c a l l e d t h e B o y d u p p e r a n d l o w e r i n d ic e s o f t h e s p a c e E [ 1 ], r e s p e c ti v e l y . W e a l w a y s h a v e
0 < a E < Z E < 1 [ 1 , p . 1 3 4 ] .
C o r o l l a r y 1 . F o r th e B o y d i n d ic e s o f r .i . s p ac e s E a n d M ( E ) t h e i n e q u a li ti e s a S <_ a M ( E ) < ] ~ M ( E ) < ~--
[3E hold .
T h e o r e m 2 ( u p p er b o u n d ). L e t E b e a n r . i. sp a c e o n I a n d l e t p = 1 / a E , w h e r e a E i s t h e B o y d
l o w e r i n d e z o f E . T h e n w e h a v e M ( E ) C L p , a n d th e e m b e d di n g co n s t a n t d o es n o t d e p e n d o n E .
C o r o l l a r y 2 . I f a E = O , t h e n M ( E ) = L o o .
I f E i s a n r . i. s p a c e o n I , t h e n t h e a s s o c i a t e d sp a c e E ' c o n s i s ts o f a ll m e a s u r a b l e f u n c t i o n s y = y ( t )
o n I s u c h t h a t
{ / 0 1 }lY llE ' = s u p x ( t ) y ( t ) d t ; I l z l l~ < 1 < c o .
T h e n o r m i n a n r .i . s p a c e E i s s a i d to b e o r d er s e m i c o n t i n u o u s i f t h e m o n o t o n e c o n v e r g e n c e x , 1" z
a l m o s t e v e r y w h e r e ( z , , z 6 E ) i m p li e s t h e r el a ti o n H z , H - + H z ]] .
C o r o l l a r y 3 . L e t E b e a n r .i . s p a c e w i t h o r d e r s e m i c o n t i n u o u s n o r m . I f B : E x E --+ E ( I x I ) a n d
B : E ' x E ' - > E ' ( I x I ) , t h e n E = L p fo r s o m e p 6 [1 , or ] .
T h e o r e m 3 (l ow e r b o u n d ) . F o r a n y r .i . s p a c e E o n I w e h a v e
M ( E ) D A ( ¢ ) ,
w h e r e ¢ ( t ) = I I ~ , I I E - + E ( 0 < t < 1 ) .H e r e t h e e m b e d d i n g c o n s t a n t d o e s n o t d e p e n d o n E .
L e t u s p a s s t o c o n c r e t e r e s u l t s . R e c a l l t h a t t h e s p a c e L p q ( 1 < p < c o , 1 < q < c o ) c o n s i s ts o f a l l
m e a s u r a b l e f u n c t i o n s z = x ( t ) o n I f o r w h i c h t h e f u n c t i o n a l
; { f ~ ( x * ( t ) t l / P ) q dr~t}1/q for 1 < q < co ,I lxllpq
s u p o < , < , ( x * ( t ) t 1 / p ) f o r q = c o ,
w h i c h i s e q u i v a l e n t t o t h e n o r m , i s f i n i t e .
T h e o r e m 4 . L e t a n r .i . sp a c e E b e 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 n Lp a n d L p , o o f o r s o m e p 6 ( 1 , c o )
( i. e ., i f a l i n e ar o p e ra t o r is b o u n d e d i n L p a n d L p , o o , t h e n i t i s c o n t i n u o u s i n E ) . T h e n M ( E ) = L p .
S i nc e L pq ( p < q _< c o ) i s a n i n t e r p o l a t i on s p a c e be t w e e n L p a n d L p ,oo [ 1, p . 142 ] , w e ob t a i n
C o r o l l a r y 4 . I f l < p < c o , p < q < c o , t he n M ( L p q ) = L p .
F o r t h e c a s e q = c o , t h e l a s t a s s e r t i o n w a s p r o v e d i n [9 ].
I n c o n c l u s io n w e p r e s e n t a n a p p l i c a t i o n o f t h e a b o v e r e s u l t s in c o n n e c t i o n w i t h t h e w e l l - k n o w n O ' N e i l
t h e o r e m o n c o n t i n u i t y o f t h e t e n s o r p r o d u c t o p e r a t o r i n t h e s p a ce s Lpq.
26 8
8/3/2019 S. V. Astashkin- On the Multiplier of a Rearrangement Invariant Space with Respect to the Tensor Product
http://slidepdf.com/reader/full/s-v-astashkin-on-the-multiplier-of-a-rearrangement-invariant-space-with 3/3
T h e o r e m 5 [8] . L e t 1 < p < oo and 1 < q , r , s < oo . The ope ra to r B f rom L pr × L p , i n to L p q ( I x I )
i s c on t inuous i f and on l y i f t h e f o l l ow ing c ond i t i ons ho ld :
( 1) m a x ( s , r ) < q ;
(2 ) 1 / p + 1 / q < 1 / s + 1 / r .
C o r o l l a r y 4 s h o w s t h a t T h e o r e m 5 i s e x a c t w i t h r e s p e c t t o t h e c l as s o f a l l r .i . s p a c e s ( a n d n o t o n l y Lpq
s p a c e s) in t h e c a s e s = p < r = q . N a m e l y , L p i s t h e m a x i m a l s p a c e a m o n g t h e r . i. s p a c e s E f o r w h i c h
B i s a b o u n d e d o p e r a t o r f r o m L p r × E i n t o L p ~ ( I × I ) .
R e f e r e n c e s
1 . S . G . K r e i n , Y u . I . P e t u n i n , a n d E . M . S e m e n o v , I n t e r p o l a t i o n o f L i n e a r O p e r a t o r s [ in R u s s i a n ] , N a u l m ,
M o s c o w , 1 9 7 8.
2 . M . M i l m a n , N o t a s M a t . , 2 0 , 1 - 1 2 8 ( 1 9 78 ) .
3 . J . L i n d e n s t r a u s s a n d L . T z a f r i r i, C l a s s i c al B a n a c h S p a c e s . V ol . 2 . F u n c t i o n S p a c e s , S p r i n g e r- V e r l a g ,
B e r l i n , 1979 .
4 . W . B . J o h n s o n , B . M a u r e y , G . S c h e c h t m a n , a n d L . T z af r ir i , S y m m e t r i c S t r u c t u r e s i n B a n a c h S p a c es ,
M e r e . A m . M a t h . S o c . , 2 1 7 , 1 - 2 9 8 ( 1 9 7 9 ) .
5 . N . L . C a r o t h e r s , I s r a e l J. M a t h . , 4 0 , N o . 3 - 4 , 2 1 7 - 2 2 8 ( 1 98 1 ) .
6 . M . M i l m a n , N o t a s M a t . , 1 3 , 1 - 7 ( 19 7 7 ).
7 . M . M i l m a n , A n a l . M a t h . , 4 , N o . 3, 2 1 5 -2 2 3 ( 1 9 7 8 ) .
8 . R . O ' N e i l , J . d ' A n a l y s e M a t h . , 2 1 , 1 2 9 - 1 4 2 (1 9 6 8 ).
9 . S . V . A s t a s h k i n , i n : S t u d i e s i n t h e T h e o r y o f F u n c t i o n s o f M a n y R e a l V a r i a b l es , Y a r o s l av l , 1 9 82 ,
p p . 3 - 1 5 .
T r a n s l a t e d b y A . I . S h t e r n
Fun ctional Ana lysis a nd Its Applications, VoL 30, No. 4, 1996
I n f in i t e -D i m e n s i o n a l N o n - G a u s s i a n A n a l y s i s
and Generalized Translation Operators
Y u . M . B e r e z a n s k y U D C 5 17 .5 15
G a u s s i a n i n f i n i t e - d i m e n s i o n a l a n a l y s i s [ 1, 2 l h a s b e e n r e c e n t l y g e n e r a li z e d t o t h e c a s e o f a n o n - G a u s s i a n
m e a s u r e , a n d o n e o f s u c h g e n e r a li z a ti o n s is b a s e d u p o n t h e r e p l a c e m e n t o f o r t h o g o n a l d e c o m p o s i t i o n s
w i t h b i o r t h o g o n a l o n e s [ 3, 4 ]. I n [ 5 -7 ] , i n t h e m o d e l o n e - d i m e n s i o n a l c a s e , i t w a s e s t a b l i s h e d t h a t t h e
a p p r o a c h o f [ 3, 4 ] c a n b e w i d e l y g e n e r a l i z ed i f o n e t a k e s t h e c h a r a c t e r s o f a n L l - h y p e r g r o u p i n s t e a d o f th e
e x p o n e n t s a n d t h e g e n e r a li z e d tr a n s l a t io n s i n s t e a d o f t h e o r d i n a r y o n e s . I n t h i s n o t e w e s h o w t h a t t h e s a m e
p r o c e d u r e c a n b e c a r r i e d o u t i n t h e i n f i n i t e - d i m e n s i o n a l s e t t i n g a n d o b t a i n r e s u l t s , f o r t h e g e n e r a l c a se ,
w h i c h g e n e r a li z e r e s u l t s o f [3 , 4 ]. N o t e t h a t h e r e a d e t a i l e d t h e o r y ( s t il l n o t c o n s t r u c t e d ) o f L l - h y p e r g r o u p s
w i t h a n o n - l o c a l l y - c o m p a c t b a s i s Q i s n o t n e e d e d a n d i t s uf fi ce s t o u s e g e n e r a l i z e d t r a n s l a t i o n o p e r a t o r s
T ~ ( z E Q ) w h i c h s at i s fy r a t h e r s i m p l e c o n d i t i o n s .
1 . L e t Q b e a s e p a r a b l e c o m p l e t e m e t r i c s p a c e o f p o i n t s z , y , . . . a n d l e t C ( Q ) b e t h e l i n e a r s p a c e
o f a ll c o m p l e x - v a l u e d l o ca l ly b o u n d e d ( i .e ., b o u n d e d o n e v e ry b a ll i n Q ) c o n t i n u o u s fu n c t i o n s o n Q . L e t
a f a m i l y T = { T ~ }~ e Q o f li n e a r o p e r a t o r s o n C ( Q ) , t h e s o - c a l l e d " g e n e r a l i z e d t r a n s l a t i o n o p e r a t o r s , " b e
g i v e n a n d h a v e t h e f o l lo w i n g p r o p e r t i e s : ( a) ( T ~ f ) ( y ) = ( T y f ) ( x ) ( x , y E Q ) f o r a n a r b i t r a r y f u n c t i o n
f e C ( Q ) ( c o m m u t a t i v i t y ) ; ( b ) t h e r e i s a p o i n t e e Q ( b a s i c i d e n t i t y e l e m e n t ) s u c h t h a t T e = i d ; ( c) f o r
a r b i t r a r y x , y E Q , t h e r e i s a b a l l W = W ~ ,y C Q s u c h t h a t , f o r a n a r b i t r a r y f u n c t i o n f E C ( Q ) , t h e
NAN U Mathematical Institute, K iev, Ukraine and M CS University, Lublin, Poland. Tran slated from FunktsionaltnyiAnaliz i Ego Prilozhenlya, Vol. 30, No. 4, pp. 61-65, October-December, 1996. Original article sub mi tted Jun e 13, 1996.
0016-2663/96/3004-0269515 .00 C)1 99 7 Plenum Publishing Corporation 2 6 9