S. V. Astashkin- Interpolation of Positive Polylinear Operators in Calderon-Lozanovskii Spaces

8/3/2019 S. V. Astashkin- Interpolation of Positive Polylinear Operators in Calderon-Lozanovskii Spaces

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Siberian M athema tical Jouraal , I /o i.38 , No. 6, 1997

I N T E R P O L A T I O N O F P O S I T I V E P O L Y L I N E A RO P E R A T O R S I N C A L D E R 6 N - L O Z A N O V S K I ~ 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 . P r e l i m i n a r i e s a n d s t a t e m e n t o f t h e r e s u l t s . L et 9 be t he c la ss o f func t io ns qo = qo(u , v )d e f i n e d f o r u _> 0 , v > 0 , p o s it i v e h o m o g e n e o u s o f d e g r e e o n e , a n d i n c r e a s in g i n e a ch a r g u m e n t . G i v e na p a i r { E 0 , E l } o f B a n a c h l a t t i c e s ( B L ) o f m e a s u r a b l e f u n c t i o n s o n a s p a c e w i t h a a - f i n i t e m e a s u r e p ,d e n o t e b y q o( E0 , E l ) 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 ) f o r w h i c h

I ~ , ( t ) l < , ~ ( x o ( t ) , x l ( t ) ) , ( 1 )w h e r e A > O , x ~ > O , : ~ e E ~ , a n d I I x ~ l l E , < 1 ( i - - 0 , 1 ) . T h e n o r m i n ~ ( E o , E ~ ) i s d e f i n e d t o b e i n fc a l c u l a t e d o v e r a l l A s a t i s fy i n g ( 1 ).T h i s c o n s t r u c t i o n f i r st a p p e a r e d i n A . P . C a l d e r d n ' s a r t i c l e [ 1 ]. T h e r e a f t e r i t s v a r i o u s a s p e c t sw e r e s t u d i e d b y a n u m b e r o f m a t h e m a t i c ia n s . I n p a r t i c u l a r , G . Y a. L o z a n o v s ki l d e m o n s t r a t e d t h a t i ng e n e r a l t h e s p a c e ~ o( E0 , E l ) i s n o t 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 E 0 a n d E 1 [ 2 ] . T h i s m e a n s t h a t n o te v e r y l i n e a r o p e r a t o r b o u n d e d i n E0 a n d E 1 is b o u n d e d i n ~o (E 0, E l ) . A t t h e s a m e t i m e , i f a l i n e a ro p e r a t o r i s p o s i t i v e t h e n t h e fo l lo w i n g i n t e r p o l a t i o n t h e o r e m i s v a li d : b o u n d e d n e s s o f s u c h o p e r a t o ri n E 0 a n d E 1 i m p l i e s i t s b o u n d e d n e s s i n ~o ( Eo , E1 ) [3].I n t h e p r e s e n t a r t i c l e , w e p r o v e a si m i l a r a s s e r t i o n f o r p o l y l i n e a r o p e r a t o r s . S t a t i n g t h e r e s u l t s,w e n e e d s o m e d e f in i t io n s a n d n o t a t io n s .G i v e n a n o n n e g a t i v e f u n c t i o n f o n ( 0 , o o ) , d e f in e it s d i l a t i o n f u n c t i o n A 4 f b y t h e e q u a l i t y

f ( s t ) ( t > o ) .~ : ( t ) = s u p 0 f ( s )S i n c e A d f i s p o l y m u l t i p l i c a t i v e , t h e r e a r e n u m b e r s

7 f = l i m l o g J ~ f ( t ) 6 f = l i m l o g J ~ f ( t )t--.0 lo g t ' t- .oo log tc a l l e d t h e l o w e r a n d u p p e r d i l a t i o n e x p o n e n t s o f f .G i v e n ~ E 4 i , p u t p ( t ) = ~ ( 1 , t ) . I t i s e a s y t o d e m o n s t r a t e t h a t 0 _< 7 p ~ 6p < 1 . H e n c e f o r t h w es u p p o s e t h a t t h e f u n c t i o n p s a t i s f i e s t h e f o l l o w i n g t w o c o n d i t i o n s :

p ( u ) p ( v ) < K p ( u v ) ,w i t h K i n d e p e n d e n t o f u > 0 a n d v > 0 , a n d

0 < T p < 6 p < 1.

( , )

( * * )S u p p o s e t h a t A is a p o ly l in e a r o p e r a to r a c t i n g b o u n d e d l y f r o m th e p r o d u c t l ' Ii ~ l x i o f B l s X ii n t o a B L Y . T h e o p e r a t o r A is c a l le d p o s i t i v e i f x l >_ 0 , . . . , x n >_ 0 i m p l i e s t h a t A ( x l , . . . , x , ) >_ O .I f X i s a B L o f m e a s u r a b l e f u n c t io n s t h e n t h e d u a l s p a c e X t c o m p r i s e s a l l m e a s u r a b l e f u n c t i o n s y ( t )

s u c h t h a t{ 'Y " E ' = s u p { / z ( t ) y ( t ) d p : ' I zI 'E < - I } < c ~"

TT h e s e c o n d d u a l i s d e n o t e d b y X " a s u s u a l .

Samara. Translated from Sibirskff Matematicheskff Zhsrnal, Vol. 38, No. 6, pp. 1211-1218, November-December,1997. Original article submitted November 8, 1995.0037-44 66/97/380 6-1047 $18.00 (~ ) 199 7 Plenu m Publ ish ing Corporat ion 1047


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T h e o r e m 1. S u p p o s e t h a t A is a b ounded p o s i t i v e p o l y l i n e a r oper ator from [I~=1 X~ in to Y0 andfrom HL 1 x~ into I:1. If a function r E r satisfies the conditions ( .) and (**) then A acts b o u n d e d l yfr om l ' I L l ~ ( X i o , X ~ ) i n t o ~ ( ~ ' , Y [ ' ) n ( Y o + Yi).

Theorem I is a consequence of, f i rst , the interpolat ion theorem for concrete weighted spaces oftwo-sided num eric sequences (Theorem 2) and, second, the possibi l i ty of describing the C alder 6n-Lozanovskii meth od in terms of the orbits of posi t ive operators (Theo rem 3).OOSuppose th at w = {w/~}k=_oo is a sequence of nonn egative num bers an d E is a BL o f numeric

OOsequences. De note by E ( w ) = E ( w k ) the BL of sequences z = (z~)k=_ oo with the fini te normI I ~ I I E ( ~ ) l l ( w k z k ) I I E .Henceforth the notat ions Ip and co are understoo d in a stan dar d way and~o'(,.,,,) I I ~ o ( I I , . , ,I ~ , ) .

T h e o r e m 2. S u p p o s e t h a t t h e hmct ion p( t ) = ~o(1, ) , ~ E # , sa t i s fi e s th e co n d i t io n s ( . ) a n d (* * ) .I f a p o s i t i v e p o l y l i n e a r o p e r a to r V a c t s b o u n d e d l y f r o m H ~ ' = I co into l l a n d f ro m I ' I~=l co(2 -k)i n t o t ( 2 - k ) t h e n i t a b o u n d e d o p e r a t o r 1 o o( '( 1, 2 - k ) ) i n t o

Recall tha t a l inear operator A is said to be a b ou nd ed o p e r at o r f r o m a B a n a c h p a i r . ~ = { X o , X1}i n to a p a i r I~ = {I~, Y1} if A is a bounded ope rator fro m X0 into I~ an d from X1 into I/1. Moreover,

l l A l l : r _ ~ - m a x { l l A l l x , - - . ~ } .i = 0 , IWe now define the orb it a nd the co-orbit of a space w ith respect to the class L()~, l~) o f al l boundedlinear operators from the pa ir X = {X0,X1} into the pair I7 = {I:0,I:1}.

Suppose th at g = {E0, E l} is a Banach pair and E is an interme diate Ban ach space for E; i .e . ,E0 N E1 C E C E0 -6 E l . Then the orb i t of E in the pair I~ = {~, Yi} is defined to be the spaceO rb g ( E , I7) of all y E Y0 -6 I:1 representable as

OOy = T j x j , ( 2 )

j=lwhere the ser ies converges in I~ -6 I~, T j e L (g , l~) , z j E E, and

OO

I I T A I E ~ . . ? I I= A I E < ~ o . ( 3 )j=lTh e norm HY][Orb is defined as the greatest low er bou nd of the sums in (3) over al l representat ions ofthe form (2) .T he space of all y E Y0 .6 I:1 w ith

HY]lCorb = s u p { ] I T y H E : I I T I I ~ _ ~ - 1 } < o ois called the co-orb i t of E in the pa i r Y and denoted by Corb (E , I~)Extremely important for in terpola t ion theory i s the possibi l i ty of descr ibing a ser ies of mostim po rtan t classes of interpo lat ion functors in terms of orbits and co-orbits (see, for instance, [4]) .Con sidering only pairs of Bls in the above defini tions and confining the exp osit ion to posit ive operators,we obta in the def init ions of O rbS (E , I7) and C or b~ (E ,Y ) . I t is c lear tha t

O r b ~ ( g , ]~) C O rbg( E , I~) , Co rb~(E , l~ ) D Co rbg(E , 17).In [5], the following result was obtained on describing the posit ive operators of the Calder6n-Lozanov-ski l me thod in te rms o f orbi ts :1048


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T h e o r e m 3 . f f qo E 9 a n d ~ = { lv , l p ( 2 - k ) } t h e n th e f o l l o w in g r e la t io n s h o ld f o r an a r b i t r a r yp a / r I~ = { Y o, r l } o f B l s :

~ ( ? ) O r b + = C t o o C ~ ' C t , 2 - k ) ) ,? ) , ( 4 )~ ( ? " ) n ( r ~ + r , ) 3 C o r b ~ ( l ~ ( ~ '( t , 2 - k ) ) , ) , ( 5)

where r = { r ~ ' ,~ ' )R E M A R K 1 . W e m a y a l s o o b t a i n t h e e m b e d d i n g ( 5) b y sl ig ht ly o d i f y i n g t h e p r o o f o f L e m m a 8 . 5.o f [ 4 1.

R E M A R K 2 . F o r l i n e a r ( n o t n e ce s sa r il y o s it i v e ) o p e r a t o r s , t h e o r e m s s i m i l a r t o T h e o r e m s I a n d 2w e r e f ir st r o v e n i n [6] f o r a n a r b i t r a r y f u n c t i o n ~ E ~ . I t is e a s y t o d e m o n s t r a t e t h a t t h e c o n d i t i o n( . ) is n e c e s s a r y i n t h e a s s e rt i o n s o f T h e o r e m s 1 a n d 2 e v e n i n t h e b i li n ea r c a s e . A t t h e s a m e t i m e , t h eq u e s t i o n r e m a i n s o p e n w h e t h e r t h e e x p o n e n t s o f t h e d i l at i o n f u n c t i o n p ( t ) = ~ ( i , t ) ( t h e c o n d i t i o n( * * ) ) m u s t b e n o nt r iv i al .

R E M A R K 3 . I n t h e p o w e r c a s e ~ (t t , ) = t t l - S v s ( 0 < s < I) , w e m a y o b t a i n T h e o r e m I a n d 2b y u s i n g t h e c o n n e c t i o n b e t w e e n t h e C a l d e r d n - L o z a n o v s k i ~ m e t h o d a n d t h e c o m p l e x i n te r p o l at i onm e t h o d t o g e t h e r w i t h t h e p o l y l i n e ar i n t e r po l a t i on t h e o r e m f o r t h e l a t t e r ( s e e [1] o r [7, p p . 1 2 5 - 1 28 ] ) .

R E M A R K 4 . I n t h e b i l i n e a r c a s e T h e o r e m I a n d 2 w e r e a n n o u n c e d i n t h e P r o c e e d i n g s o f t h eI n t e r na t i o n al C o n f e r e n c e d e d i c a t e d t o t h e 9 0 t h a n n i v e r s a r y o f A c a d e m i c i a n S . M . N i ko l ls k i~ [8].

2 . P r o o f s .P R O O F O F T H E O R E M 2 . W e u s e t h e i d e a o f r e p r e s e n t i n g a n o p e r a t o r a s t h e s u m o f " d i a g o n a lope r a t o r s by ana l ogy t o t he l i nea r ca se i n [ 9 ] ( s ee a l so [ 4 , p . 464 ] ) .U s i n g s ta n d a r d a r g u m e n t s , w e c a n e q u i v a le n t l y r e s t a t e t h e a s s e r ti o n o f t h e t h e o r e m a s f ollo w s :i f V i s a b o u n d e d p o s i t i v e p o l y l i n e a r o p e r a t o r a c t i n g f r o m l'I ~f fitc 0( p( 2k )) i n t o l l ( p ( 2 k ) ) a n d f r o m

l 'I ~ = l c o (p ( 2 k ) 2 - k ) i n t o l l ( p ( 2 k ) 2 - k ) w i t h t h e r e s p e c t i v e n o r m s C o a n d C 1 t h e n V a c t s b o u n d e d l y f r o mI ' [~=l loo into 11, where p ( t ) = qo (1 ,t ) a s above . D eno t e I l V l l = m a x ( C 0 , C ~ ) .

T h e o p e r a t o r V i s d e t e r m i n e d b y t h e f o l lo w i n g s e q u e n c e o f i n f in i t e n u m e r i c " m a t r i c e s " :~ k t~ k k= ( ~ , . . . , , ~ ) , ~ , , . . . , ~ > 0 ,_ ( p ~ , . . . , w ) e z ~ , k e z .

9 O 0I n d e e d , s u p p o s e t h a t { e ,} i = _o o is t h e s t a n d a r d b a s i s f o r t h e s p a c e co a n d a i = ~ o = _ ~ a ) e i ( 1 < i < rt)i s a c o m p a c t l y - s u p p o r t e d s e q u en c e . T h e n t h e l i n e a r i ty o f V i n e a c h a r g u m e n t i m p l i e s t h a t

v ( , : , a " ) ( k ) k = - ~ , b ~ O 0 O 0E - - Ep l = - - o O p t t = - - r

1 n k~ p l 9 . . t t p a ~ p l , . . . , p n ,

w h e r e v k - - ( V ( e p t , e l ~ ) ) k .P l , . . . , P n " " " ~E n u m e r a t e t h e s y s t e m o f th e R ~ t e m a c h e r f u n c t i o n s o n [ 0,1 ] s o t h a t i t c o n s t i tu t e a n i n f in i t e tw o -s i ded sequ enc e ( h j( t) } ~~ w i t h h o ( t ) = 1 . G i v en m E Z an d t E [0 ,11 , de f i ne t he seq uen ce o f" m a t r i c e s "

9 kv ~ , , = h k _ ~ , f t ) (h _ , , . . . . . , ~ ( t ) ~ , , . . . . . , ~ ) , ( p l , . . . , p n ) ~ Z ~ , k z z .E a c h o f t h e s e s e q u e n c e s a ls o d e t e r m i n e s a b o u n d e d p o l y l i n e a r o p e r a t o r f r o m 1 - I ~ ' = l c 0 ( p ( 2 k ) ) i n tol l ( p ( 2k ) ) and f r om l ' I ~ ' = l c0 ( p ( 2k ) 2- k ) i n to l l ( p (2 k ) 2 - k ) w i t h t h e r e s p e c t i v e n o r m s n o t e x c e e d i n g I lV ll-A v e r a g i n g o v e r a l l t E [0 , 1] w i t h r e s p e c t t o t h e L e b e s g u e m e a s u r e , w e i n f e r t h a t a s i m i l a r p r o p e r t yi s e n j o y e d b y t h e o p e r a t o r V m d e t e r m i n e d b y t h e f o l lo w i n g s e q u e n c e o f " d ia g o n a l m a t r i c e s " :

v . , = % , .....n , ~ , k - ~ - , 1 . .. . . _ , ) . ( ~ . p n ) ~ z ~ ,1049


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w h e r e / f i , j i s t h e K r o n e c k e r s y m b o l . T h e r e f o r e , i n v i e w o f t h e c o n d i t i o n ( . ) , f o r m > 0 w e o b t a i n

I Iv i i > _ I I V - I I ~ , 0 ( , , ( 2 , . ) ) . _ . , , ( , , o , ~ ) )i=1

- I[ (p (2 ;'z p(2k))] . p ( 2 ~ ) v w v " v " ' k - r a - v t. . . . . . . . . e " - ' ) I] H e 0 . . .hi = l

tl> K - " p ( 2 " ) II V . , II I .i ~ . . , ,i---1

B y n o n n e g a t i v i t y o f v ~ .....~ , t h e l a s t in e q u a l i t y i s e q u i v a l e n t t oO0 O0

Ek--- -- -oo Pl =--OO

CO p , , .. ., r ~ _ 1 , k -m - p , . . . . . p . - , < I l V l l K npn- - l =- -O0

o r I l V l l K n< ( r . > o ) . ( 6 )I IV I IH , * * _ , , , - _ p ( 2 m ) -i = 1

E s t i m a t i n g f r o m b e lo w t h e n o r m o f t h e o p e r a t o r V m f r o m H ~ f l c 0 ( p ( 2 k ) 2 - k ) i n t o / 1 ( p ( 2 k ) 2 - k ) i n t h ec a s e o f m < 0 , w e o b t a i n 2 " I I V I I K "I I V ' l l f i , . . _ , 1 < p ( 2 , , , ) ( 7 )l li = lS i n c e V = ~ = - o o V ,,,, t h e in e q u a l it ie s (f i) a n d ( 7 ) im p l y t h a t

OOI I V I I H * - - * , m = - ' - - ' ,

i----1 i=1w h e r e

C 1 " - E ~ - < O Qm f f i O P

b y ( * * ) . C o n s e q u e n t l y , V : H~'=l/OO -, 11 a n d T h e o r e m 2 i s p r o v e n .P R O O F O F TH E O R E M 1 . S u p p o s e t h a t A i s a b o u n d e d p o s i t iv e p o l y l in e a r o p e r a t o r f r o m H : = I X ~

i n t o 1 ~ a n d f r o m H ~ = I x ~ i n to Y 1 w i t h t h e r e s p e c ti v e n o r m s C 0 a n d C 1 a n d p u t BAH = m a x ( C 0 , C l ) .I f z i E ~ ( X ~ , X ~ ) t h e n z i E Orb+ o. ( /oo(~o*(1 ,2- t ) ) , -~ i ) w i th .~ i . .. { X i o , X ~ } ( i = 1 , 2 , . . . , n ) i n

v i e w o f ( 4 ) . F i r s t s u p p o s e t h a t t h e r e a x e T i > 0 , T i E L ( i ' ~ , - ~ i ) , a n d a i E / o o ( ~ o * ( 1 , 2 - k ) ) f o r w h i c h= i = T i n i ( i = 1 , 2 , . . . , n ) . A s a b o v e , ~ = { t p , /p ( 2 - k ) } .

G i v e n a b o u n d e d p o s i t iv e l i n e a r o p e r a t o r Q f r o m t h e p a i r I 7 = { I ~ , Y 1 } i n t o t h e p a i r ~ , d e f in e t h eo p e r a t o r V : V(~l,...,~n) = QA(TI~I,...,Tn~n).n n - kT h e n V i s p o s i t i v e , p o l y l i n e a x , a n d a c t s b o u n d e d l y f r o m l " I i f x l o o in to Ix an d f rom I 'I i=~loo(2 ) in to

/ l ( 2 - k ) . M o r e o v e r ,

[ I V I I = m a x ( l lV l ] f i z z _ h , l l V l l H , . ( ~ _ , ) _ , . ( 2 _ , ) - ) < - I I Q I I I I A I IH I I T ~ I I ,~ = ,i=1 i=l

1 0 5 0


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I I Q I I = I I Q I l e _ r ~ , I I T i l l = I I T d l r ~ _ x , .B y T h e o r e m 2 , V a c t s b o u n d e d l y f r o m l ' l$ =x lo o( ~* (l , - k ) ) i n t o 1 1( ~ *( i , - k ) ) w i t h a n o r m n o t e x -c e e d i n g C1 [[ V ][.

S u p p o s e t h a t y = A ( x , , . . . , x , ) = A ( T x a ~ , . . . , T , , a ,) . B y t h e d e f i n i ti o n o f C o rb + , w e h a v eI l Y l IC o , b + = s u p { l l Q u l l ~ , ~ : I I Q I I - 1 } = s u p { l l V ( a x , . . . , . . ) l l ~ a : I I Q I I - 1 }

n- < c L l ' I l l a , l l , , , ~ o s u p { l l V l l : I 1 r 1 } < C x l l A I l I I I I T , I I I l a , l l ~ , , o o ,i=1 i=1w here I1" I1~, , s tan ds for the no rm in the sp ace Iv(~0*(1 ,2-k) ) . Hence , owing to the em be dd ing (5) , wein f e r t ha t t he f o l l owing i nequa l i t y ho ld s w i th s ome C2 > 0 :

I l y l l ~ - C 2 1 1 A I I I I I I T d l I l a , l l ~ , o o , ( 8 )i=l

where I1" I1 ~ i s t h e no r m in t h e s pace r ( Y~ ', YI ") N ( I/ 0 -I- ~ ) .I n ,t he gene r a l c a s ei ixi = Z Ti a i ,

1=19 " i Ioo(~o*(1,2-/~)) an d, f inal ly,he r e t he s e r i e s converges in X ~ + X l , T ~ e L ( ~ o o , g ) , a e

( 9 )

O0 J J ( i 1 , 2 , . . . , , ) .l i T , I l r | l l a ~ l b , o o < o o =i=1

F o r a n a r b i t r a x y c o ll e ct i on P l , - . - P n ) E ~ w e p u t= ,"pP 1 ~,P lY p ,, . . . ,p , A ( ~ I . . 1 , . . . , T ~ a ~ ) 9

Then , t he l i neax i t y o f T~ and t he po ly l i neax i t y o f A y i e ldO O O O

y = A ( ; r , , , . . . , = , ) = y ~ ' " '" Z Yvb.. ., ,~"pl=l pn=l

A p p l y i n g ( 8 ) t o yp~ ....p . , w e o b t a i nn

I ly , ~ . . . . . i l ~ _ ~ C ~ I I A I I I I I I T P I I I I ~ P I b . ~ .i=lwhe nc e

OO OO OO OO u

p 1 = 1 p r im 1 p l = l ~ = I i = 1

I l I I a iC ~ l l a I I , 1 1 ~ , ~ 9i=l "=1051


6/7

T a k i n g t h e g r e a t e s t l o w e r b o u n d s o n t h e r i g h t - h a n d s i d e o v e r t h e r e p r e s e n t a t i o n s ( 9 ) , f r o m t h ed e f i n it i o n o f t h e n o r m i n O r b + w e d e ri v e

n

I I A ( x x , . . . , ~ , ) l l ~ _ _ C 2 1 1 A I l~ l l ~ l lo , b + .i=!

H e n c e , A a c t s b o u n d ~ U y fr o m r I L - l ~ ( X ~ , x ~ ) i n t o ~ ( Y g , Y ~ ' ) n ( ~ + Y ~ ) a n d I I A I I~ - < C 3 1 1 A I I n v ie wo f ( 4 ) . T h e t h e o r e m i s p r o v e n .A p p l i c a t i o n s . H e r e w e j u s t s ck e tc h s o m e p o s si b le a p p l ic a t io n s o f T h e o r e m 1 .P O L Y L IN E A R I N T ERP O L A T IO N IN Lp -S P A CE S. S u p p o s e t h a t w - - w ( t ) i s a p o s i t i v e m e a s u r a b l ef u n c t i o n o n a s p a c e T w i t h a m e a s u r e p . T h e n , a s u s u a l , L p ( w ) c o n s is t s o f a l l m e a s u r a b l e f u n c t i o n sz = z ( t ) o n T f o r w h i c h w ( t ) z ( t ) G Lp an d [[zl[LAw ) = [[wz[[/ ;~ .

S i n c e ~ ( L p ( w ~ L p ( ~ * ( w ~ [4 , p . 4 5 9 ], T h e o r e m 1 y i e ld sT h e o r e m 4 . / f ~ G 9 satisf ies the condit ions ( * ) a n d ( * * ) a n d a posi t ive po lyl inear opera tor Aa c t s b o u n d ed ly f r o m 1-[~'=,L, w ~ i n to L , ( ~0 ) ~ d f r o m l -[~ '=~L,(w~) i n t o L , Cv l ) ( 1 < p ,q _~ o o ) t h e n

A a c t s b o u n d ed ly f rom r i ~ f l L , ( ~ ' ( w ~ i n to L , ( ~ o ' ( v 0 , v , ) ) .R E M A R K 5 . I n t h e c a s e o f p = q = o o a s i m i l a r r e s u l t w a s o b t a i n e d i n [ 1 0] f o r a r b i t r a r y b i l i n e a r

( n o t n e c e s s a r i l y p o s i t i v e ) o p e r a t o r s .POLYLINEAR INTERPOLATION IN ORLICZ S P A C E S . S u p p o s e t h a t N ( t ) i s a n i n c r e a s i n g c o n v e xf u n c t i o n o n [ 0 , o o ) a n d N ( 0 ) = 0 . T h e O r l i cz sp a c e L ~v 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 = z ( t )o n T s u c h t h a t

/ N ( l z ( t ) l / u ) d p 0 w i t h t h e n o r m, , . , , - - , n , { . > 0

TL * L *o n s i d e r a p a i r { jV o, N , } o f 0 r l i c z sp a ce s. G i v e n ~ ~ ~ , d e n o t e b y N ( t ) t h e f u n c t i o n s u c h t h a t

N - ' ( t ) = ~ ( ~ o ~ ( t ) , ~ i - ~ ( t ) ) ,H ,w h e r e N - l ( t ) i s t h e i n v e r s e f u n c t i o n o f N . S i n c e ~ ( L ~ r , , L ~ , ) - L~v [4 , p p . 4 6 0 - 4 61 ] a n d ( L ~ r ) - L N ,

f r o m T h e o r e m 1 w e o b t a i n t h e f ol lo w i n g n a t u r a l p o l y ~ n e a r i n t e r p o l a t i o n t h e o r e m f o r O r l i c z s p a c e s:T h e o r e m 5 . Lf ~ E ~ satisfies the conditions ( . ) , , a n d ( * * ) and a po lyl inear posi t ive opera torA a c t s bou nded ly f rom rI i 1L*~ in to L * and f rom rI i 1L*M~ n to L* t h e n A a c t s b o u n d ed ly f r o m- - -- - i / r -~ ' 1[I~=,L~, into L'R, , .her e

a . l C t ) = ~ ( N . 1 C t ) , M . l ( t ) ) C i = 1 , . . . , - ) , a - l C t ) = ~ ( R 0 - 1 C t) ,a ~ lC t ) ).R E M A R K 6 . I n [1 0] , a s i m i l a r r e s u lt w a s o b t a i n e d f o r r e a r r a n g e m e n t i n v a r i a n t M a r c i n k i e w i c zs p a s m [1 1, p . 1 5 2 ] a n d a r b i t r a r y b i l i n e a r ( n o t n e c e s s a r i ly p o s i t i v e ) o p e r a t o r s .CONVOLUTION O P E R A T O R S A N D T E N S O R P R O D U C TS . D e n o t e b y S t h e b i l i n e a r c o n v o l u t i o n

o p e r a t o r f o r f u n c t i o n s d e f i n e d o n R m :/ x ( t - , )y ( s ) d ~ , t E R ~( z , y ) ( t )Q/

T h e o r e m 6 . I f t h e o p era to r S a c t s b o u n d e d l y f r o m X ~ x X ~ i n t o Y 0 a n d f r o m X 1 x X21 into Y1( X a n d 1~ a r e B l s of funct ions def ined o n R " ) and the funct ion ~ E ~ sa t i s f ies the condi t ions (* )a n d ( * * ) t h e n S a c t s b o u n d ed ly f ro m ~ ( X ] , X I ) ~ ( X ~ , X ~ ) i nt o ~ ( Y g , Y ? ) n (Y o + Y 1 ).1052


7/7

I n s t u d y o f s o m e p r o b l e m s c o n n e c t e d w i t h t h e g e o m e t r i c p r o p e r t i e s o f r e a r r a n g e m e n t i n v a r i a n ts pace s [ 11 ] an i m po r t an t r o le i s p l ayed by t he t ens o r p r od uc t ope r a t o r= (s )y C t) ,

w h e r e z - x(s ) and y - y( t ) a re m easu rable fun c t ions o n [0, 1] ( see , for ins tan ce , [12 , 13]) .T h e o r e m 7. If the operator B acts boundedly from X~ x X2o into Yo a n d from X~ x X21 into

I ~ ( X ~ a n d I ~ a r e r e a r r a n g e m e n t invariant spaces on [0,1] and on [0,1] x [0,1]) and the funct ione ~ satisfies the c o n d i ti o n s ( . ) a n d ( * * ), then B acts boundedly f rom ~ ( X ~ , X I ) ~ ( X ~ , X [ ) i n t o

n +Th e au t h o r exp r e s s e s h is g r a t i t ud e t o V . I . Ovch i nn i kov f o r f r u i tf u l d is cus s ions .

R e f e r e n c e s

1 . A . P. C a l d e r o n , " I n t e r m e d i a t e s p ac e s a n d i n t e r p o l a ti o n . T h e c o m p l e x m e t h o d , " S t u d i a M a t h .Appl . , 24 , 113-190 (1964) .2 . G . Ya . Lozanovs k i l , "A r emar k on a c e r t a i n i n t e r po l a t i on t heo r em o f Ca l de r on , " Funk t s i ona l .Anal . i Pr i lozhen . , 6 , No. 4 , 89-90 (1972) .3 . E . I . Be rezhnoY, " I n t e r po l a t i on o f l i nea r and com pac t ope r a t o r s i n t he s pace s ~ ( X0 , X1) , " i n :Q u a l i ta t i v e a n d A p p r o x i m a t e M e t h o d s f o r S t u d y i n g O p e r a t o r E q u a t i o n s [ in R u s s ia n ] , Y a r o sl av s k .Un i v . , Y a r os lav l~, 1980, pp. 19-2 9.4 . V . I. Ovch i nn i kov , "Th e me t h od o f o r b i t s i n i n t e r po l a t i on t heo r y , " M a t h . Rep . , 1 , No . 2 , 349 - 515(1984).5 . V .V . Vodop~yanov, "O r b i t s and co - o r b i ts o f a pos i t ive in t e r po l a t i on f o r Ban ach i dea l s tr uc t u r e s , "I zv . Vys s h . Uchebn . Zaved . M a t . , No . 3 , 76 - 78 ( 1989) .6 . V . I . Ovch i nn i kov , " I n t e r po l a t i on t heo r ems t ha t a r i s e f r om Gr o t hend i eck ' s i nequa l i t y , " Funk t -s iona l . Anal . i Pr i lozhen . , 10 , No. 4 , 45-54 (1976) .7 . J . Be r gh and J . LSf s t r Sm, I n t e r po l a t i on Spaces. I n t r od uc t i on [ Rus s ian t r ans l a t i on ] , M i r , M os cow(1980).8 . S . V . As t a s hk i n , "The b i l i nea r i n t e r po l a t i on t heo r em f o r s pace s o f t he Ca l de r on - Lozanovs k i lm e t h o d , " i n : A b s t r a c t s: T h e I n t e r n a ti o n a l C o n f er e n ce o n F u n c t io n S p a c e s , A p p r o x i m a t e T h e o r y ,and No n l i nea r Ana l ys i s , M oscow, 1995 , pp . 319 - 320 .9 . S . J ans on , "M i n i ma l and m ax i m a l m e t hod s o f i n t e r po l a t i on , " J . F unc t . An a l . , 44 , 50 - -73 ( 1981) .10 . S . V . As t a s hk i n and Yu . E . Ki m , " i n t e r po l a t i on o f b i l i nea r ope r a t o r s i n M ar c i nk i ew i cz ' s s pace s ,"M at . Za m etki , 60 , N o. 4 , 483--494 (1996).11 . S . G . Kr e~n , Yu . I. Pe t un i n , and E . M . Sem~nov , I n t e r po l a t i on o f L i nea r O pe r a t o r s [ in Rus s ian ] ,Nauka , Moscow (1978) .12 . N .L . C a r o t he r s , " l ~ea r r angemen t i nva r i an t s ubs paces o f Lo r en t z f unc t i on s pace s ," I s r ae l J . M a t h . ,40 , No. 3-4 , 217-228 (1981) .13 . S . V . As t a s hk i n and M . Sh . Br ave r man , ~A s ubs pace o f a r e a r r angem en t i nva r i an t s pace wh i ch isgene r a t ed by a 1L 'Ldemache r s y s t em w i t h vec t o r coe f fi c ien t s, " in : O pe r a t o r Equ a t i ons i n F unc t i onSpaces [ in Russ ian] , Voronezh . Univ . , Voronezh , 1986, pp . 3-10 .

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S. V. Astashkin- Interpolation of Positive Polylinear Operators in Calderon-Lozanovskii Spaces

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