S. V. Astashkin- One Property of Functors of the Real Interpolation Method

8/3/2019 S. V. Astashkin- One Property of Functors of the Real Interpolation Method

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O N E P R O P E R T Y O F F U N C T O R S O F T H E R E A L I N T E R P O L A T I O N M E T H O DS . V . A s t a s h k i n

w h e r e u / ~ X o ~ X 1.in (I).

I . I n t r o d u c t i o nW e r e c a l l s o m e d e f i n i t i o n s f r o m i n t e r p o l a t l o n t h e o r y o f l i n e a r o p e r a t o r s ( f or d e ta i l ssee [i] or [2]).A B a n a c h s p a c e X i s s a i d t o h e i n t e r m e d i a t e f o r a B a n a c h p a i r X = ( X 0 , X I) i f X o ~ X , C

X C X o q - X I ( h e n c e f o r t h , t h e e m b e d d i n g s a r e c o n t i n u o u s ) ; h e r e , t h e c o l l e c t i o n ( X 0, X ,, X )i s c a l l e d a t r i p l e . W e s a y t h a t a t r i p l e ( X o , X ,, X ) i s i n t e r p o l a t i v e r e l a t i v e t o a t r i p l e(Yo, Y,, Y),if for ea ch lin ear op er at or T: X i - + Y i (i = 0, 1) T (X) c Y . If X ffi Y, X is sai d t obe a n i nte rpo lat ive space bet wee n the pai rs (X0, X,) and (Y0, Yt) (if, mor eove r, Xl = Yi (i = 0,i ), t h e n X is a n i n t e r p o l a t i v e s p a c e b e t w e e n t h e s p a c e s X 0 a n d X I ) . I n p a r t i c u l a r , f o r e a c hi n t e r p o l a t i v e f u n c t o r ( i . f .) ( s ee [ I] o r [ 2] ) F a n d a r b i t r a r y B a n a c h p a i r s ~ = ( X 0, X l) a n dF = (Yo, YI) the tr iple (X0, XI, F (~)) If, in a ddit ion, for e ach line ar o per ato r (Y0, Y,, F (~)

H T ~ F ( ~ - - F ~ ) < m ax (I] r [ [ x i ~ y t ) ,i=O, 1 ' '

t h e n F i s s a i d t o b e a n e x a c t i . f .F o r a B a n a c h p a i r ( X0 , X I ) a n d 0 < s , t < o o , w e d e f i n e P e e t r e ' s ~ "- a n d ~ - , f u n c ti o n a l s [ 3 ]:

Y ~" s , t . z ; X o , X , ) = i n f i s ~ X o f ix . + t ]1z l ~ , } ( z E X o + X , ) ;x ~ X

@ , t , z ; X o , X , ) - - -- - m a x { s . [[ x [ I x ., t . ~ z H x , } ( x E X o ~ X 1 )I f s = i , t h e s e f u n c t i o n a l s a r e d e n o t e d b y , ~ (t, x; X0, X I) an d ~ (t, z; Xo, XI).

L e t E b e a B a n a c h i d e a l s p a c e ( B IS ) ( e . g ., s e e [ 4 ]) o f t w o - s i d e d n u m b e r s e q u e n c e sa = (a/)j~__~. We d eno te b y (X0, X,)~ the s et of all x ~ X o ~ - X I , for wh ic h the nor m []z H =[[(~(2, z; X0, XI))jH~; is fin ite ; by (Xo, X,)~ the set of all x ~ X o q - X l , a d m i t t i n g a r e p r e s e n t a -t i o n

z = ~ = _ ~ u ~ ( c o n v e ~ e n c e i n X o + X,), (I)The no rm in (X0, XI) is de fi ne d as inf9( (2 , ufi X o , X , ) ) / I I E over all (uj)

F o r a B I S o f s e q u e n c e s G a n d a f u n c t i o n f , w e d e n o t e b y G ( f ) t h e s p a c e w i t h t h e n o r mH (ak ) fIG(t) [[ a k" ] (2~)[) a. If I= (m ax (l,I/t)) E C It rain (I, I/t)), t he n th e ma p (X0, X,) ~ (X0, XI )~( r e s p e c t i v e l y , ( X o , X , ) ~ - ~ ( X o , X x ) ~ ) d e t e r m i n e s a n e x a c t i .f . ; t h o i r c o l l e c t i o n i s c a l l e d t h e~ m e t h o d ( r e s p e c t iv e l y , t h e ~ - m e t h o d) o f i n t e r p o l a t i o n. A n i m p o rt a n t s p e c i a l c a s e ( e .g .,s e e [ i ] ) i n v o l v e s t h e s p a c e s :

~ (0 0 < I , 1 < p < ~ ) .X 0 , X , ) 0 , = ( X 0 , X 0 9 < , - 0 = ( X 0 , X l )~ , ~, - 0~ 0 w e d e n o t e b y t R t h e s p a c e o f r e a l n u m b e r s w i t h t h e n o r m l la l~ s = t . ] a I.I f F i s a n e x a c t i . f . , t h e n i t s c h a r a c t e r i s t i c f u n c t i o n i s d e f i n e d b y t h e i s o m e t r i c e q u a l l t y :

F (n, 1 / t R) ---- t l g (t)) R. (4 )LEMMA 1. Suppose tha t F is an exac t i . f . wi th th e char ac te r is t ic func t ion R and , fo r

so me I < Po, P* < oo, F (It, ~'), l~, ~*~) = E. Th en the se qu en ce s (ll ll~) a n d ( ~' ( 2 ) j a r e e q u i v -a l e n t .

~ r o o f . F o r e a c h k ~ Z w e d e f i n e o p e r a t o r s T ~ ( a j ) = a ~ a n d S ~ a = a . e ~ ( a ~ R ' . S i n c e

a n d ( s e e ( 4 ))II T a 1 1 , ,( ,7 , )_ ( t7 % % ~ 1

F ( (1 ~ ' ( 2 b ) R , ( f i ' ( 2 D ) R ) = ( ~ ' ( 2 ~ ) ) . s ,w e d e d u c e , b y t h e e x a c t n e s s o f th e i . f. , t h a t

wher e C1 > 0 does not dep end on k~Z .S i m i l a r l y , b y v i r t u e o f t h e f a c t t h a t

a n d b y r e l a t i o n s h i p ( 5) , w e o b t a i n :

( 5 )

(6)

V S ~ ] 1 ( , ~ , (? ) ) . _ E < c , , ( 7 )w h e r e C2 > 0 a l s o d o e s n o t d e p e n d o n k ~_ Z . T h e s t a t e m e n t o f t h e l e m ma f o l l o w s f r o m ( 6 ) a n d(7).

W e r e c a l l s o m e n o t a t i o n f r o m [ 7 ]. S u p p o s e t h e s h i f t o p e r a t o r P~. { a j ) = ( a~ +a) j~ __~ i s c o n -t i n u o u s i n t h e B I S E f o r a l l k ~ Z T h e n t h e r e e x i s t n u m b e r s

I n IIP k ~ E - - EP E= l i m VE ~ li m I n I I P ~ i E ~ Ek - - , - - ~ k ' k - * ~ k

I n t h e n e x t t h r e e s t a t e m e n t s f 0 , f l a n d E a r e l i k e t h o s e i n T h e o r e m i .LE MM A 2. If (If e~ IIm~ -~ (g-1 (2;.))~, th en ?g~o* > 0 an d 6g?, | (g is a p os it iv e fu nc ti on on

I t s u f f i c e s t o s h o w t h a t f o r s o m e C > 0 a n d a l l k ~ Z. . , , g , ? ( 2 ~ ) < c . II P - ~ I I E (1 o ) ~ E ( !. ) ( 8 )

'#gT, (2~) ~ C II P ~ ! IE O ~I )~E O ;~ 9 (9)

( 0 , ~ ) ) .P r o o f .

a n d

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I n d e e d , t h e n , b y t h e f a c t t h a t E i s i n t e r p o l a t i v e a n d b y T h e o r e m I i n [ 7 ] , y g , ~ , / > ~ E ( / , ) > 0a n d 6 g & . ~ VE~7~ t .

F o r e a c h k = 0 , 1 , . w e f i n d J k f o r w h i c h

g (2 - ~ + J ~ ) ./ o 2 ~ ) > + * ' ~ ' g , o I ( 2 -~ ) .1 o ( 2 - ~ ' & ) 'r ( 2 ~ 5 "T h e n , b y h y p o t h e s e s ,

w h e n c e9 t 1 0 ( 2 ~ + ~ ), t Io(2 ok) >~ 2Cx " ~ g ~ o * ( 2 - ~ ) ~ " g g ~ o * ( 2 - ~ ) ~ e - ~ + i ~ ( m 'I P-:-~+& i e u . ) -- - - Jl % h E u . ~ ~ > C ~ ~ ( 2~ ,~ ) . g ( 2 - k + ~ )

J /g to ~ (2 -~) . < 2 .6 ~ , . s u pm ~ Z O P -~ ' ,I E (, . > 2 . ~ . U - ~ U ~ , . ~- ~ U a 'era IE(&)and, theref ore, (8) is proved. The proo f of (9) is similar.

T h e f o l l o w i n g t w o s t a t e m e n t s a r e g i v e n w i t h o u t p r o o f s . I n t h e c a s e w h e n E i s Z p w i t h" w e i g h t , " t h e f o r m e r h a s b e e n p r o v e d i n [ 6 ] ( s e e T h e o r e m 1 3 ); t h e e x t e n s i o n t o t h e g e n e r a lc a s e p r e s e n t s n o d i f f i c u lt i e s . T h e p r o o f o f t he l a t t e r st a t e m e n t g e n e r a l i z i n g t h e f u n d a-m e n t a l l e m m a of i n t e r p o l a t i o n t h e o r y I s e s s e n t i a l l y i d e n t i c a l t o t h e p r o o f o f t h a t l e m m aits elf (e.g., see [I, p. 62]).

P r o p o s i t i o n I . F o r e a c h B a n a c h p a i r X = ( X o, X x) h o l d t h e e q u a l i t i e s

a n d

LEMMA 3.II = il.(~ ) = II (.~"Uo (2~), h (2~), x; X o, X ,))~ I[~.

S u p p o s e t h a t f o r s o m e x ~ X o -~- X xl i r a [ra in ( /?* (2~))- ~g' ( /o (2~), l* (2~), x; Xo , XO ] = O.

k-*i - r i - - - -O,1( 1 o )

T h e n f o r e a c h e > 0 t h er e e x i s t s a r e p r e s e n t a t i o n z = ~ , ~ = _ u s u c h t ha t f o r k ~ Z( f0 2 5 , , 2 ~ ), ~ ; 0 , x , ) < ( y + 8 ) x ( 1 o ( 2 ~) , ( 2 ~) . ; x 0 , x , ) ,

w h e r e y ~ 3 i s a u n i v e r s a l c o n s t a n t .W e c o n t i n u e t h e p r o o f o f T h e o r e m i. W e w l l l f i r s t s h o w th a t e a c h B a n a c h p a l r ( X 0 , X l )

a n d e a c h z ~ H ( X 0 , X l ) s a t i s f y ( I0 ).L e t g H b e t h e c h a r a c t e r i s t i c f u n c t i o n o f th e l . f . H . T h e n , b y ( 3 ) , [8 , p . 4 8] , a n d

als o [9, p. 23],

s o , b y P r o p o s i t i o n I , f or e a c h z ~ H ( X e, X ,) t h e r e e x i s t s C > 0 s u c h t h a tX (/o (2~), h (2~), x; X 0, X,) < C./ ,, (2~). ( 1 1 )

On the othe r hand , (3) and L emm a I impl y th e eq uiv ale nce of the se que nces (R e~ ~) and (/~ (2~)),w h e n c e , b y L e m m a 2 a n d L e m m a 3 in [ 7 ] , 0 < ? g, ~ 6 g, < I. T h u s , i n p a r t i c u l a r ,l i m [ m i n ( t , t / t ) g . ( t ) ] - -- -- 0 . ( 1 2 )t - - ~t - ~ a o

F ro m ( 1 1 ) , ( 1 2 ) , a n d t h e f a c t t h a t t h e r e l a t i o n / , .f o 1 t s i n c r e a s i n g f r o m 0 t o | f o l l o w s t h ev a l i d i t y o f (1 0 ) f o r x E H ( X 0 ,X 1 ).

E v e r y t h i n g i s n o w r e a d y f o r t h e p r o o f o f t h e e q u a l i t y G = H . S u p p o s e a l i n e a r o p e r a t o r) )T: l, (~') ~ l~ (~') (i = 0, I) a nd a ~ E Sinc e E is in te rp ol at ive be tw ee n l, (~*) an d I. (~'),)T a ~ _ . E , and, thus, l l ( l , ( / - * ) ) ~ E . Since G is the least i.f. po sse ssi ng thls !)roperty, we

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o b t a i n : G < H . C o n v e r s e l y , l e t x ~ H (X 0, X I ~ . T h e n ( 1 0 ) h o l d s a n d , b y L em ma 3 , f o r e a c he > 0 t h e r e e x i s t s a r e p r e s e n t a t i o n x = ~ , : = _ = uh. s u c h t h a t f o r k ~ Z

(/0 (2k) , /1 (2~), u , ; Xo, X 0 < (? + e ) X ( /0 (2 ' ) , h (2 ' ) , x ; Xo, X 0.T h e r e f o r e , b y P ro p o s i t i o n 1 , x ~ G (X0 , XI ) a n d I[ x [[G(X,, X0 ~ (Y J r e ) [[ X ][UCX,.X,).

T h u s , G = H . W e d e n o t e t h e t . f . G b y F t . , q . E . B y d e f i n i t i o n , F / , , h . g p o s s e s s e s t h e r e -q u i r e d p r o p e r t i e s . W e w i l l p r o v e i t s u n i q u e n e s s . S u p p o s e t h a t f o r so m e i . f . F , F , ( l (/-x)) =F l ( l | = E . T h e n , s i n c e G a n d H a r e e x t r e m a l , G ~ F l a nd F I S H , w h e n c e F I = F I , , I , E .T h e o re m 1 i s p r o v e d .

T h e f u n c t o r F / , , h , E c o n s t r u c t e d i n T he o re m 1 d e p e n d s , i n g e n e r a l , o n t h r e e p a r a m e t e r s :f 0 , f l , a n d E . U n d er c e r t a i n a s s u m p t i o n s o n E , t h i s d e p e n d e n c e i s s u b s t a n t i a l l y s i m p l i f i e d .R e c a l l t h a t a B I S o f s e q u e n c e s E is s a i d t o b e s y m m e t r i c i f f or e a c h i n j e c t l v e m a p n : Z - ~ Z ,a = (aj~-~_-~ g a n d

1[a n [ ! E = ] a li e ( a n = ( a .n ( n ) j ~ = - * * ) .TH EO REM 2 . S u p p o s e t h a t E i s a s y m m e t r i c s p a c e o f s e q u e n c e s , / 0, ]1, h , , h a, g a r e q u a s i -

c o n c a v e f u n c t i o n s o n ( 0, ~ ) , th e f u n c t i o n s / i .~ a n d h l . h ~ ar e in c r e a s i n g , ?h.1o-, 0, ?h,.n0-, 0a n d 0 < ? g ~ 6g < ~ . Th e n FI , " h , E "~ Fn , , h , , E h , w h er e E l -- -- E ( i f1) an d Eh = E (h~l) .

T w o l e m m a s a r e n e c e s s a r y f o r t h e p r o o f , t h e f i r s t o f w h i c h i s p r o v e d i n a s t a n d a r dm a n n e r .

L E M M A 4 . L e t ( g0 , E ,, E ) b e a t r i p l e o f B I S o f s e q u e n c e s , v a p o s i t i v e f u n c t i o n o n ( 0,~) Th en O E = ,~E(~)~'z(~), w h e r e ~ = (Eo , El) E (u) = (E 0 (~), El (u)).

L E M H A 5 . S u p p o s e t h a t f u n c t i o n s [o, ],, g a n d w =/ ,' if 0* s a t i s f y t h e h y p o t h e s e s o f T h e o r e m2 . F o r A > 0 , B > 0 , w e d e f i n e a s e t V ( A , B ) ~ Z b y

v CA, B) = {~ : A < g ( ~ ( 2~ )) < B ) .T h e n f o r e a c h C ~ ~ 0 t h e r e e x i s t s C ~ ~ 0 s u c h t h a t B . A - * ~ C , im pl ie s card F (A, B) ~< C,.

P r o o f . S i n c e ~ > 0 a n d ? g > 0 , T O l i m w ( t)~ --l im i = l i m g ( t ) = l i m !t-.0+ ,-.~ ~ (t) ,_~+ , -- ~ - ~ = ~ O. T h e r e f o r e ,f o r a n y A > 0 a n d B > 0 t h e r e e x i s t n = 0 , 1 2, . . a n d m = 0 , I , . . , s B c h t h a t

g (w (2~)) A < g (w (2 +')),g (w (2 , ,+ , - - ,) ) < n < g (w (2 - - - ) ) .

C l e a r l y , e a r d V ( A , B ) ~ < m q- l , a n d i f B A - * ~ < C I , t h e ng (w (2n+ra)) Bg ( ~ ( 2 ~) < 4 - 7 - < 4 C ~ .

B y t h e h y p o t h e s e s o f t h e l em m a , f o r s om e r > 0 a n d k ~ _ Z

So~' (w (2x ') ) ~ g (og w ( lh) w (2~+1))

g (w (2 t ) ) g (w (2~

w h e n c e2 = 8 < g ( w ( 2 "+ ~ ) ) < 4 . G ,g (w (2") )

c a r d V ( A , B ) ~ < m + 1 ~ e - l . l o g 2 ( 4 C I ) - k t - -= - C ~ .P r o o f o f T h e o r e m 2 . B y T h e o r e m 3 i n [ 7 ] , E f i s i n t e r p o l a t i v e b e t w e e n l~ ( ] - ) a n d l = ( ] - !)

( r e s p e c t i v e l y , E h b e t w e e n ll (h ~) a n d l ' ~ ( h - ~ ) ) . F u r t h e r m o r e , b y T h e o r e m 1 a n d L ernm a 4 , F I . , t , . E !i s t h e l e a s t a mo ng i . f . F s a t i s f y i n g t h e c o n d i t i o nF ( l l " ( g ( f l . f o l ) ) , l l ( g ( / I " ] - - 0 1 )/ v l ~ 1 ) ) ~ E .

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A similar statement is true for F ~ , ~ , , E ~ . So for the proof of the theorem it suffice s to showt h e v a l i d i t y o f t h e i n c l u s i o n

Fn..~,. a ( l , ( g ( / t. ~ * ) ) , l,(g(h'l~x.~* )) ~ E , (13)and then use the extremal ity of the i.f. F& , ~,,~! and eq ual sta tus of fi an d hi.

Since ?n,.~,>O, we have

lira ~ = lira ~=0,t-o+ o ( ) ,_~ ~()so for each k ~ Z there exists

9 f . . h ( 2 t ) " ~] , , = m , . l l . 1 0 - j - < -ho (2~ )N e x t , i t f o l l o w s f r o m t h e d e f i n i t i o n o f J k a n d q u a s i c o n c a v i t y o f t h e f u n c t i o n s h z . ~ x a n d gthat

and

L et N 1 = c a r d { k : ] k = J } .d e n o t e

W e w i l l p r o v e t ha t N = s u p N I < oo ..~ez

A - = g ~ } .To this end, we fix j and

Then, if k is such that Jk = J, then, b y vir tue of (14),9 h ( 2 ~ ) '~---A ~ 0 does not depend on A and, theref ore, on ]E Z. Thus, N < ~.Now, w e defin e for e ach a = (a~ )~ E a seque nce a'-----{a~)~:

{ m a x i ~ t ~ [ ] E V ,~ t ~ J k = JO, j~V ,

w h e r e V i s t h e s e t o f a l l j ~ Z , f o r w h i c h J k = J f o r s o m e k ~ Z . S i n c e E i s a s y m m e t r i cs p a c e , a ' = (a } ) ~ . -E a n d Ila'n ~


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I n e x a c t l y t h e s a m e w a y b u t u s i n g t h e r e l a t i o n s h i p ( 15 ) o n e p r o v e s t h a tI I T z i l , , , r ~ = r ,Tn" ' < z ' /v l lz l l', ( ~ ' (~ ) ) "

S i n c e t h e i . f . F h , , h , , E h i s t h e o r b i t o f E r e l a t i v e t o t h e p a i r ( l , ( g ( h , . t ~ l ) ) , l a ( , g (ka'h~ ( s e eT h e o r e m 1 a n d L e m m a 4 ) , t h e r e l a t i o n s h i p ( 13 ) a n d w i t h i t, a s h a s a l r e a d y b e e n n o t e d , a l s oT h e o r e m 2 a re p r o v e d .

C O R O L L A R Y I , S u p p o s e t h a t E i s a s y m m e t r i c B I S , f 0 , f l , a n d g a r e q u a s i c o n c a v e f u n c -tio ns o n (0, ~), /i.]~0 is inc rea sin g, ~t,.to ~ 0, 0 < ~g ~ 8e < I. Then for ea ch Ban ach p air( X 0 , X l )

f ( X o, X O = ( X o , a t " ~OE 0, 0 < ? h- ,< 6 ~ < I; E i s a s y ~ a e t r i c B I S o f s e q u e n c e s . T h e n t h e s e t- h 'l 0 9 B ~ , ~ = ( . ,I , (p*),. . . / * * ( f - a ) } is su ff ic ie nt for the i.f. (', at .~_,).~

3 . C a s e o f S y m m e t r i c F u n c t i o n a l S p a c e sW e d e n o t e b y A { ~ t h e B a n a c h p a i r o f L o r e n t z s p a c e s ( A (]0), ( ]D ), a n d b y M - ~ ) t h e p a i r

o f M a r c i n k i e w i c z s p a c e s ( M (f,), M ( f D ) .T H E O R E M 4 . S u p p o s e t h a t f 0 , f l , a n d g a r e c o n c a v e f u n c t i o n s o n {0, c o) , [ i .~ i s i n c r e a s -

t n g ; ] , h . t g , ? 0 , 0 < y t i f q < | ( i = 0 , t ) , 0 < ? , . ~ < 6 e < t , E i s a s y m m e t r i c B IS o f s e q u e n c e s . T h e n(A (j), M Or)) is a su ff ic ie nt set of the i.f. (., ")~-9-

I n o r d e r t o p r o v e t h e t h e o r e m w e w i l l n e e dP r o p o s i t i o n 2 . S u p p o s e _ t h a t B I S o f s e q u e n c e s E 0 a n d E l a r e i n t e r p o l a t i v e b e t w e e n Z m

a n d l* * ( t / t ) a n d f o r e a c h ( a ~ ) ~ E 0 U E ,l ira a~-----0.

/ r

The n th e e qua li ty (L,, ~)~.----(L~, ~)E, imp lie s E0 = El.P r o o [ , S i n c e t h e c o n d i t i o n s a r e s y m m e t r i c r e l a t i v e t o E0 a n d El , i t s u f f i c e s t o s h o w

that E0 C E,.If ( e l ) ~ E o, t h e n t h e s e q u e n c e ( ~Z k) w h e r e

a~ = sup [ rain ( t , 2u-0 9 a j l]~_ za l s o b e l o n g s t o E0 ( s e e [ 1 0 ] o r [ 1 1 ] ) , s o l ir a a ~ . = 0 . W e d e n o t e

S i nc e f or a l l 0 < t 1 < G < o oa ( t O < a ( t ~ ) .t.____L < , , ( t , )

t8 tl 'w e d e d u c e t h a t , b y [ 2 ] ( p . 6 9 ) , a ( t ) i s e q u i v a l e n t t o i t s l e a s t c o n c a v e m a j o r a l a t a ( t ) ,l i r a a ( t ) - - - - l i r a a ~ = 0 .t-*O+ k--~-**p. 1 0 8 ] ,

a n dT h e r e f o r e , ~ ( t ) i s a b s o l u t e l y c o n t i n u o u s o n [ 0 , = ) a n d , t h u s , b y [ 2,

a ( t ) = I 'o a ' ( s ) = x ( t , a ' ; L , , L . ) .

7 3 1


8/8

L ~h u s , I ,~[Z~,a ';Li , L~)) ' ----(u~), w h e n c e a ' ( ! j ( L , , L . ) ~ ( L , , ~ ) ~ a n d (a~)~--=E,. S i n c e l a k i -. < a~ , w ec o n c l u d e t h a t ( a ~ ) ~ E , T h e p r o p o s i t i o n i s p ro v e d .

P r o o f o f T h e o r e m 4. T h u s , s u p p o s e t h a t a n i . f. F s a t i s f i e sF (A (0) = (A (f))~{~-,),F ( ~ ) ) = ( ~ ) ~

By [7, Theo rem 3], the spac es I (J~') and l~ (J~) are int erp ola tiv e bet we en the pa irs (l~, I (I/t)a n d (l~. ~ (|It~. F u r t h e r m o r e ( s e e [ 8 ~) ,

A ( I ~ ) ---- (L ,, L =)l ,(17, ) , M (f0 -~-- (L ,, L~)l |S o , b y r e i t e r a t i o n a r g u m e n t s ( s e e [ 1 2 , C o r o l l a r y 3 ] o r [ 10 , C o r o l l a r y 1 4 . 1 ] ) a n d b y C o r o l -l a r y 2 , ~ e o b t a i n :

( L t , L . ) r ( ~ _ ~ ) = ( / a , L ~ ) p < ~ 9 ) ----- L , , ~ )E (, ~t .A d i r e c t v e r i f i c a t i o n s h o w s t h a t t h e s p a c e s F (l (~)), F (l= (~')) and E (~i) sat is fy the hy -(IA))o t h e s e s o f P r o p o s i t i o n 2 a n d , t h u s , F (l, = F ( l ~ ( E l ) ) = E ( ~ i ) . T h e r e f o r e , b y T h e o r e m 3 ,F = ( ' , " )~ < ,; n. T h e o r e m 4 i s p r o v e d .

I n c o n c l u s i o n , t h e a u t h o r e x p r e s s e s h i s d e e p g r a t i t u d e t o E . M . S e m e n o v a n d V . l .O v c h i n n i k o v f o r v a l u a b l e d i s c u s s i o n s .

L I T E R A T U R E C I T E DI . I . B e r g a n d I . L ~ f s t r o m , I n t e r p o l a t i o n S p a c e s . I n t r o d u c t i o n [ R u s s i a n t r a n s l a t i o n ] ,

M i r , M o s c o w ( 1 9 8 0 ) .2 . 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 [ i n

R u s s i a n ] , N a u k a , M o s c o w ( 1 9 7 8 ) .3 . J . P e e t r e , " A t h e o r y o f i n t e r p o l a t i o n o f n o r m e d s p a c e s , " N o t e s M a t h . , 3 9 , 1 - 8 6 ( 19 6 8 ) .4 . L . V . K a n t o r o v i c h a n d G . P . A k i l o v , F u n c t i o n a l A n a l y s i s [ i n R u s s i a n ] , N a u k a , M o s c o w

(1977).5 . N . A r ~ n s z a j n a n d g . G a g l i a r d o , " I n t e r p o l a t i o n s p a c e s a n d i n t e r p o l a t i o n m e t h o d s , " A n n.

Math. Pure Appl. , 2__4, No. 2, [[3-19 0 (1964).6 . S. J a n s o n, ' ~ i n l m a l a n d m a x i m a l m e t h o d s o f i n t e r p o l a t i o n , " J . F u n c t i o n a l A n a l y s i s , 4 4 ,

No. I, 50-7 3 (1981).7 . S . V . A s t a s h k l n , " A d e s c r i p t i o n o f 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 ( l, (~ ), / ,( ~ )) a n d l ~ (~ ) ,

l,(~'))," Mat. Zam etk l, 3_~5, No. 4, 49 7- 50 3 (1984) .8. V. I. Dmlt riev , S. G. Kreln, a nd V. I. Ove hin nik ov, " F u n d a m e n t a l s o f t h e i n t e r p o l a t i o nt h e o r y o f l i n e a r o p e r a t o r s , " i n: G e o m e t r y o f L i n e a r S p a c e s an d O p e r a t o r T h e o r y [ i nR u s s i a n ] , Y a r o s l a v l ( 1 9 7 7 ) , p p . 3 1 - 7 4.

9. I. U. Ase krit ova , "On the 3~-functlonal o f t h e p a l r ( ~ r ( ~) , ~ | , " i n : S t u d i e s i n t h eT 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 i r i a h l e s [ i n R u s s i a n ] , Y a r o s l a v l ( 1 9 80 ) , p p. 3 - 32 .

1 0. Y u . A . B r u d n y i a n d N . Y a . K r u g l y a k , " R e a l i n t e r p o l a t i o n f u n c t o r s , " D e p . V I N I T I , N o .2 6 2 0 - 8 1 .

1 1. P . N i l s s o n , " R e i t e r a t i o n t h e o r e m s f o r r e a l i n t e r p o l a t i o n a n d a p p r o x i m a t i o n s p a c e s , "L u n d , L T H ( 1 9 8 2 ) .

1 2. V . I. D m i t r i e v a n d V. I . O v c h i n n i k o v , " O n i n t e r p o l a t i o n i n s p a c e s o f t h e r e a l m e t h o d , "D o k l . A k a d . N a u k S S S R , 2 4 6 , No . 4 , 7 9 4 - 7 9 7 ( 1 9 7 9 ).

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