S. V. Astashkin- Disjointly Strictly Singular Inclusions of Symmetric Spaces

8/3/2019 S. V. Astashkin- Disjointly Strictly Singular Inclusions of Symmetric Spaces

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Mathematic al Notes, Vol. 65, No. I, 1999

D i s j o i n t l y S t r i c t l y S i n g u l a r I n c l u s i o n s o f Symmetric SpacesS . V . A s t a s h k i n U D C 5 1 7 . 9 8 2 . 2 7

A B S TR A C T . I n t h i s p a p e r , t h e d i s j o i n t s tr i c t s i n g u l a r it y o f i n c l u s i o n s o f s y m m e t r i c s p a c e s o f f u n c t i o n s o n a ni n t e r v a l i s c o n s i d e r e d . A c o n d i t i o n f o r th e p r e s e n c e o f a " g a p " b e t w e e n s p a c e s s u f f ic i e n t f o r t h e in c l u s i o n o fo n e o f t h e s e s p a c e s i n t o t h e o t h e r t o b e d i sj o i n tl y s t r ic t ly s i n g u l a r is f o u n d . T h e c o n d i t i o n i s s t a t e d i n t e r m s o ff u n d a m e n t a l f u n c t i o n s o f s p a c e s a n d i s e x a c t i n a c e r t a in s en s e . I n p a r a l le l , n e c e s s a r y a n d s u f f i c ie n t c o n d i t i o n sf o r a n i n c l u s i o n o f L o r e n t z s p a c e s t o b e d i s j o in t l y s t r ic t l y s i n g u l a r ( a n d s i m i l a r c o n d i t i o n s f o r M a r c i n k i e w i c zs p a c e s) a r e o b t a i n e d a n d c e r t a i n o t h e r a s s e r ti o n s a re p r ov e d .K E Y W O R D S : B a n a c h s p a c e , d i s j o in t l y s tr i c t ly s i n g u l a r o p e r a t o r s , i n c l u s i o n o p e r a t o r s , i n c l u s i o n s o f s y m m e t r i cs p a c e s , f u n d a m e n t a l f u n c t i o n , L o r e n t z s p a c e s, M a r c i n ld e w i c z sp a c e s , O r li c z s p a c e s .

I n t r o d u c t i o nR e c a l l t h a t a b o u n d e d l in e a r o p e r a t o r T f r o m a B a n a c h s p a c e X i n t o a B a n a c h s p a c e Y i s c a l le ds t r i c t l y s ingu lar ( o r a K a t o o p e r a t o r ) i f X d o e s n o t c o n t a in a n i n f i n i te - d i m e n s i o n a l s u b s p a c e Z s u c h t h a t

t h e r e s t r i c t i o n o f T t o Z i s a n i s o m o r p h i s m .I n r e c e n t d e c a d e s t h e c l a s s o f s t r ic t l y s i n g u l ar o p e r a t o r s h a s b e e n e x t e n s i v e l y s t u d i e d ( s ee t h e r e f e r e n c e s

c i t e d i n , e . g. , t h e m o n o g r a p h [1 ]). O n e o f t h e h i s to r ic a l ly f i rs t r e s u lt s i m p o r t a n t f o r o u r p u r p o s e s i s t h eG r o t h e n d i e c k t h e o r e m o n t h e s t r i c t s i n g u la r i ty o f t h e i d e n t i ty in o n o p e r a t o r f r o m L ~ ( ~ , # ) i n t o L p ( l ~ , # ) ,w h e r e 1 < p < o o an d /~ is a p r o b ab i l i t y m eas u r e o n 12 ( s ee [2 ] o r [ 3 , T h eo r e m 5 .2 ]) . H o w ev e r , a s a r u le ,i n o n s o f s y m m e t r i c s p a c e s ( t h e d e f in i ti o n i s g i ve n b e lo w ) a r e n o t s t r i c t l y s i n g u l a r b e c a u s e o f t h e e x i s t e n c eo f " t h r o u g h " s u b s p a c e s ( s u c h a s t h e s u b s p a c e g e n e r at e d b y th e R a d e m a c h e r f u n c t i o n s [ 4]). I n p a r t b e c a u s eo f t h i s , t h e c l o s e n o t i o n o f d is j o i n t l y s t r i c tl y s i n g u l a r o p e r a t o r w a s i n t r o d u c e d i n 1 9 8 9 [ 5].

A h o u n d e d l i n e a r o p e r a t o r T f r o m a B a n a c h l a t ti c e X i n t o a B a n a c h s p a c e Y i s c a l le d d i s jo in t l ys t r i c t l y s ingu lar (o r h a s t h e D S S p r o p er t y ) i f t h e r e ex i s t s n o s e q u en ce o f n o n ze r o d i s j o i n t v ec t o r s { ~n }n ~_ _li n X s u ch t h a t t h e r e s t r i c t i o n o f T t o th e i r c l o s ed l i n ea r h u ll [ x,~] i s an i s o m o r p h i s m .

C l e a r ly , a n y s t r i c t l y s i n g u l a r o p e r a t o r is a D S S o p e r a t o r . A s im p l e e x a m p l e s h o w s t h a t t h e c o n v e r s ei s n o t t r u e . F o r i n s t a n c e , t h e i d e n t i t y i n o n o p e r a t o r I : L p [ O , 1] -+ Lq[O,1] (1 _< q < p < oo ) ha st h e D S S p r o p e r t y , b e ca u s e t h e c l o s ed l i n ea r h u ll i n L ~ o f d i s j o i n t f u n c t i o n s z ,~ E L ~[0 , 1 ] i s i s o m o r p h i cto tT (1 O ) t h e d i s t r ibu t ion funct ion o f z . T w o f u n c t io n s z ( t ) a n d y ( t ) a r e ca l l edequimeasurable i f n . ( r ) = nu(~" for r > O.T r a n s l a t e d f r o m Maternaticheskie Zametki, V o l . 6 5 , N o . 1 , p p . 3 - 1 4 , J a n u a r y , 1 9 9 9 .O r i g i n a l a r ti c l e s u b m i t t e d A p r i l 1 8 , 1 9 96 .

0 0 0 1 - 4 3 4 6 / 9 9 / 6 5 1 2 - 0 0 0 3 $ 2 2 .0 0 ( ~) 19 9 9 K l u w er A c a d e m i c / P l e n u m P u b l i s h e r s


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R e c a l l t h a t t h e B a n a c h s p a c e E o f m e a s u r a b l e f u n c t i o n s o n [ 0, 1 ] i s c a l le d s y m m e t r i c ( b r i e f ly is a n S S )i f t h e f o l l o w i n g c o n d i t i o n s h o l d :

(1) i f y 9 E a n d I z ( t ) l _< I v ( t ) t , t h e n z 9 E a n d l l ~ l l -< Ilyll;( 2 ) i f v 9 E a n d fu n c t i o n s z ( t ) a n d v ( t ) a r e e q u i m e a s u r a b l e , t h e n z 9 E a n d I 1~ 1 1 = I l Y ll -

T h e f u n d a m e n t a l f u n c t i o n o f a n S S E i s d e f i n e d b y r E ( t ) = I ] x r w h e r e , a s u s u a l ,1 , t E U ,

o, t ~ u .T h e f u n c t i o n r E ( t ) i s qua s i c onc a ve on ( 0 , 1 ] [ 8, p . 1371 , i . e ., i t i s nonn e ga t ive , i nc r e a se s , a nd f ( t ) / td e c r e a s e s . A s is k n o w n ( s e e , e . g ., [ 8 , p . 7 0 ] ) , s u c h a f u n c t i o n 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 n t .T h r o u g h o u t , G d e n o t e s t h e c l a ss o f a ll p o s i t i v e i n c r e a s i n g f u n c t i o n s c o n c a v e o n ( 0 , 1 ] .

A n i m p o r t a n t e x a m p l e o f a n S S i s a n O r li c z s p a c e. L e t N ( t ) b e a n 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 , o o ) s u c h t h a t N ( 0 ) = 0 a n d N ( o o ) = o o . T h e O r l ic z s p a c e L N c o n s i s t s o f a l l f u n c t i o n s z = z ( t )m e a s u r a b l e o n [0 , 1] a n d s u c h t h a t

\ u ]f o r s o m e u > 0 ; t h e n o r m o f t h i s s p a c e i s] , z l l : i n f { u > O : / T N ( l Z ( - - t u ) - - ~ l ) d l ~ < _ l } .

D i r e c t c a l c u la t i o n s h o w s t h a t t h e f u n d a m e n t a l f u n c t i o n o f t h e s p a c e L N is f N ( t ) = 1 / N - ~ ( 1 / t )( N - l ( u ) i s t h e i n v e r s e o f g ( u ) ) [9].I n [ 5 ], t h e f o l lo w i n g d i s j o i n t s t r i c t s i n g u l a r i t y t h e o r e m f o r i n c lu s i o n s o f L N i n t o L M i s p r ove d .

T h e o r e m . If LN C LM , the n the followin g conditions are equivalent:(1 ) the inclusion I : LN --+ LM is a D S S operator;(2 ) f o r any n = l , 2 , . . , and lC > O, t here ez i st 1 O suchthat

n n

ci N (t m ) >_ 1C E ci M (t zl ) for t >_ 1.i=1 i=1

L e t u s s h o w t h a t c o n d i t i o n ( 2 ) f o ll o w s f r o m t h e r e l a ti o nElm fM (t ), - ~ o f N ( t- -- -S = 0 ,

w h e r e f N a n d f M a r e t h e f u n d a m e n t a l f u n c t i o n s o f t h e r e s p e c t i v e O rl ic z s p a c es . I n d e e d , t h is r e l a ti o ni m p l i e s t h a t N -1 ( t ) < h M -~ ( t ) f o r a n a r b i t r a r y p o s i t i v e h < 1 a n d t > t o a n d , s i n c e N ( t ) i s c onve x ,M (t ) < N (h t ) _ M -~ (to ) . T h e r e f o r e ,

l lm M(t ) _ O ,t ~ r N ( t )a n d c o n d i t i o n ( 2 ) h o l d s .

Q u i t e n a t u r a l l y , t h i s o b s e r v a t i o n l e a d s u s t o t h e f o l l o w i n g g e n e r a l p r o b l e m .S u p p o s e t h a t f u n c t i o n s ~a E G a n d r E G s a t i s fy t h e c o n d i t i o n

9 r( A ) ~ - - ~ = 0


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and are, respectively, the funda me nta l functions of symm etri c spaces E and F such tha t E C F. Doesthe identity inclusion oper ator I: E --+ F have the DSS prope rty?

In what follows, we show that the answer to this question for "classical" symmetric spaces (such as theLorentz and Marcinkiewicz spaces) an d for Orlicz spaces is positive. Moreover, it is so for an inclusion ofa Lorentz space into an arbitrary SS (and, vice versa, of an arbitrary SS into a Marcinkiewicz space).

However, in the general case, this is not true: this paper contains an example of two symm etric spaces Eand F such tha t E C F and their fundam ent al functions satisfy condition (A), but the opera tor I : E -+ Fis not DSS.

At the same time , it is possible to state a condition on fund ame nta l function s stronger than (A) underwhich the answer to the stated question is positive for all symmetric spaces. First, recall the definition ofthe dilation of a function.

For a positive functio n f on (0, 1], its d i la t io n M / ( t ) is defined asM s ( t ) = s u p 0 < s < r a i n 1 ,

Since A ~ l ( t ) is semimultiplicative, there exist numbers7 f : l i r a l n j ~ f ( t ) and ~ f - - lim h a f l 4 f ( t )t-~0 In t t--.o~ In t '

which are called, respectively, the l o w e r an d u p p e r d i l a t i o n s of the function f . If ~ E G, then we have0 < 7~ _ < ~ o .

The definition of lower dilation readily implies that condition (A) follows from (B). The converse, of course,is not true: it suffices to take for ~ and ~ functions differing by a logarith mic factor (see also the proofof Theorem 3).

We shall show tha t, i f (B) holds, the n the operat or I: E ---> F will be a DSS ope rat or for arb itr arysymmetric spaces E and F, E C F, with fundame ntal functions ~ and ~ , respectively. This resultgeneralizes and simultaneously refines a similar theorem for the Orlicz spaces proved in [7]. In parallel,we shall show that condition (A) is necessary and sufficient for the identity inclusion operator from oneLorentz space into another to have the DSS property (and a similar assertion for Maxcinkiewicz spaces).These results also supplement the theorem for Orlicz spaces proved in [5] and cited above.

w Th e inclus ions A(~) C F and E C M ( O )For ~ E G, t he Loren tz space A(~) consists of all functions x -- x ( s ) m e a s u r a b l e on [0, 1] and such

that / o i d (s) 0 imply tha t r _ 0 such that

[ [ z , ~ l L h ( ~ ) < C~llz~llF for n = 1 , 2 , . . . . (1)


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B y c o n d i t i o n ( A ) , f o r a n y 0 < e < 1 , t h e r e e x is t s a n h > 0 s u c h t h a tr < ~ ( t ) ( 2 )

f o r a l l p o s i t i v e t < h . C h o o s e N s o t h a t , f o r n > N , # ( g = ) < h , w h e r e g = = { t E [ 0, 1 1 : z , ~ ( t ) # 0 } .T h e s u b s e t o f f i n i t e - v a l u e d f u n c t i o n s i s d e n s e i n a n y L o r e n t z s p a c e o n [ 0, 1 ] [ 8, p . 1 49 ]. T h e r e f o r e , f o re a c h n > N , t h e r e e x i s t s a f u n c t i o n

mr s

= = = ~ = a n d # ( ~ ) < h ,, t( t ) = a k x e ~ , w h e r e a ~ > 0 , e 1 D e2 D - . . D m . ,k= l

f o r w h i c hm a x ( l l z . - w l l A c ~ ) , I 1 = , - y = I I A ( ~ ) ) < ~ m i n ( l l z . l l ^ ( ~ ) , I 1 = = 1 1 ~ ( ~ ) ) .

H e n c e I 1 = .1 1 ^ ( ~ ) - I l y . I I A ( ~ ) _ < ~ 1 1 = . 1 1 ^ ( ~ , a n d [ 8, p . 1601 a n d ( 2 ) im p ly tha t1 i l Y ~ l l ^ ( ~ ) = iI I ~ , , l l v _ < I 1 1 ~ , . , . 1 1 ^ ( ~ ) _ < i - ~ 1 - ~ : mnr - I - ~

k - - - - I

I n a d d i t i o n , b y ( 3 ) , I l y = l l ^ c ~ ) < ( 1 + ~ ) t 1 = = 1 1 ^ ~ ) - T h u sI I ~ - I I F - 0 s a t is f y in g

e(1 + ~)~ > ~

71%n I l Y , , l l ^ c , . ) -k= l

( 3 )

c o n t r a d i c t s ( 1 ) . [ ]C o r o l l a r y 1 . S u p p o s e t h a t ~ o E G , r E G , a n d 4 ? ( t) 0 s u c h t h a tI 1 ~ : , ,1 1 ^ ( ~ ) _ < c 2 1 1 z , , . l l ^ ( ~ ) f o r n = 1 , 2 , . . . . ( 4 )

P r o o f . T h e i m p l i c a t i o n s ( 1) == v ( 2 ) a n d ( 2 ) ===~ 3 ) f o ll o w f r o m T h e o r e m 1 a n d t h e d e f i n i t i o n o f a D S So p e r a t o r , r e s p e c ti v e l y .

S u p p o s e t h a t c o n d i t i o n ( A ) d o e s n o t h o l d , i .e ., th a th" rm ~ > 0 ., - . . o , p C t )

T h e n t h e r e e x is t a s e q u e n c e { t k } C ( 0 , 1 ] a n d a c o n s t a n t C 2 > 0 s u c h t h a t w e h a v e ~ = a t,~


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T h e o r e m 2 . L e t t h e f u n c t i o n s T 9 G an d ~b 9 G sa t i s fy cond i t ion ( A ) . I f E i s a n S S o n [0, 1] w i t hf u n d a m e n t a l f u n c t i o n ~o ( t) , t h en E C M( ~b ) a n d t h e i d en t i t y i n c l u s i o n 1 : E --+ M ( r i s a D S S opera tor .P r o o f . S i n c e a n y M a r c i n k ie w i c z s p a c e i s m a x i m a l a m o n g a ll s y m m e t r i c s p ac e s w i t h t h e s a m e fu n d a m e n t a lf u n c t i o n [ 8, p . 1 6 2 ], w e h av e E C M ( ~b ) . B y co n d i t i o n ( A ) , r t ~) (k = 1 , 2 , . . . ) .S i nc e t h e f u n c t i o n s x k a r e d i s jo i n t , w e c a n a s s u m e t h a t t k ~ 0 . T h e r e f o r e ,

I I ~ ' ~ I I M ( ~ > - - ( ~ ( a ) ) - 1 f o ' ~ z ~ ( s )d s > - -v ( t ~ )2 W ( t k )B y ( A ) , I l x k l l M ( ~ ) - + o o a s k - + o o , w hi c h c on t ra d i c t s c ond i t i on (5 ) .

T h i s c o m p l e t e s th e p r o o f o f T h e o r e m 2 . [ ]C o r o l l a r y 2 . Sup pos e tha t ~o 9 G , r 9 G , a n d r < C l i o ( t ) f o r t 9 (0 , 1] . T h e f o l l o w i n g c o n d i ti o n sare equ iva len t :

(1 ) (A) h o l d s ;(2 ) t h e i n c l u s i o n I : M ( ~ ) --+ M( ~b ) i s a D S S opera tor ;(3 ) t h e r e ex i s t n o s eq u en ce o f d i s j o i n t f u n c t i o n s X n a n d n o co n s t a n t C 2 > 0 s u ch t h a t

I I ~ n l lM ~ ) ~ C 2 I I~ , ~ IIM ( ~ ,) f o r n = 1,2, . . . .

T h e p r o o f o f C o r o l l a r y 2 is s i m i l a r t o t h e p r o o f o f C o r o l la r y 1 .C o r o l l a r y 3 . F o r a n a r b i t r a r y S S E # L 1 o n [0 , 1] , the inclu sion I : E --+ L1 is a D S S opera tor .P r o o f . F i r s t , L x = M ( 1 ) , a n d a n a r b i t r a r y S S E i s e m b e d d e d i n L1 [8 , p . 1 2 4] . I f r E ( t ) = ~0( t ) , thent h e f u n c t i o n t / T ( t ) i n c r ea s e s , b ec au s e So i s co n cav e . T h e r e f o r e , co n d i t i o n ( A ) is v i o l a t e d i f an d o n l y if~0(t ) ~ t ( i .e . , i f an d on ly i f C lt < qo(t) 0 an d C 2 > 0 ) , an d E = L 1 . I t r em a i n s t oa p p l y T h e o r e m 2 . [ ]R e m a r k 1 . T h e a s s e r t i o n o f C o r o l l a r y 3 w a s p r o v e d in [1 0] i n a d if fe r e n t w a y .R e m a r k 2 . A r g u i n g a s i n t h e p r o o f o f C o r o ll a ry 3 a n d a p p ly i n g T h e o r e m 1 , w e c a n re a d i ly s h o w t h a tt h e i n c l u s i o n I : L o~ --+ E i s a D S S o p e r a t o r fo r an y S S E ~ L ~ . M o r eo v e r , it i s s h o w n i n [1 1]t h a t t h i s o p e r a t o r i s e v e n s t r i c t l y s i n g u la r . T h i s ge n e ra l iz e s th e G r o t h e n d i e c k t h e o r e m m e n t i o n e d i n t h ei n t r o d u c t i o n .

I n t h e n e x t s e c t i o n w e s h o w t h a t , g e n e ra l ly , c o n d i t i o n ( A ) is n o t s u f f ic i e n t f o r t h e i n c l u s io n o f a n S Sw i t h f u n d a m e n t a l f u n c t i o n q0 i n t o a n S S w i t h f u n d a m e n t a l f u n c t i o n ~b t o h a v e t h e D S S p r o p e r t y .


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w A n e x a m p l e o f s y m m e t r i c s p a c e s E a n d F s u c h t h a t E C F a n d t h e i rf u n d a m e n t a l f u n c t i o n s s a t i s f y c o n d i t i o n ( A ), b u t I : E --+ F i s n o t a D S S o p e r a t o r

T h e o r e m 3 . T h e r e e z i s t t w o s y m m e t r i c s p ac e s E a n d F o n [0, 1] w i t h f u n d a m e n t a l f u n c t i o n s ~ a n d ~br e s pe c t iv e l y , s u c h t h a t E C F , ~ a n d 42 s a t i s f y c o n d i t i o n ( A ) , a n d t h e o p e r a t o r I : E ~ F d o e s n o t ha v eth e D S S p r o p e r t y .P r o o f . L e t t h e S S E b e t h e M a r c i n k ie w i c z s p a c e M ( r w i t h ~ ( t ) = t / 4 2 ( t ) , w h e r e

42(t) t 1/2 9 1 / 24 O < t < l .tog~. ~ ,I t i s r e a d i l y v e r i f ie d t h a t 42 i s a n 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 ] a n d V r = t ic = 1 / 2 . T h e r e f o r e , [ 12 ,C h a p . 6 , H i n t t o E x . 6 ],

I l Z l l M ( ~ ) ~ s u p { z * ( t ) r (6 )0 < t < l

L e t u s d e f in e t h e s p a c e F . W e p u t b k = ( k + 2 ) - ~ / 2 2 k /2 a n d z k ( t ) = b k x ( 0 , 2 - k ] ( t ) a n d d e f i n e a s e q u e n c eo f n u m b e r s n o = 1 < n l < - - - < r tm < - - - b y s e t t i n g

,~-1 1 < 1} (7)n ,n +x = m a x n = l , 2 , . - . : k + 2 -k ~ r t ma n d a s e q u e n c e o f f u n c t i o n s w , ~ = w , n ( t ) b y s e t t i n g

w , ~ ( t ) = m a x z k ( t ) , m = O , 1 , . . . .n,~


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T h e r e f o r e , ( 8 ) a n d t h e d e f i n i t i o n o f F i m p l y t h a t I 1 1 / r < _ 4 , w h i c h p r o v e s ( 9 ) .N e x t , w e p u t D , ~ = ( 2 . . . . + 1 , 2 - ~ , ,, ] a n d

n m + i - - 1

V m ( t ) = W r n ( t ) X D . , ( t ) = E b k x ( 2 - k - l , 2 - k ] ( t) f o r m = 0 , 1 , . . . .k~ n ~ r ,

T h e f u n c t i o n s v ,~ a r e d i s j o i n t . L e t u s s h o w t h a t t h e n o r m s o f E a n d F a r e e q u i v a le n t o n t h e ir l i ne a rhu l l .

S u p p o s e t h a tv ( t ) = ~ a , , ~ v m ( t ) .

m-~OW i t h o u t l o ss o f g e n e r a l it y , w e c a n a s s u m e t h a t a m > O . C o n s i d e r

w ( t ) = m =0 < m < r

T h e f u n c t i o n w ( t ) m o n o t o n i c a l l y d e c r e a s e s o n ( 0 , 1 ] , a n d v ( t ) < w ( t ) . T h e r e f o r e , b y ( 6 ) ,

[ IV N E < l l w i l E < C m a x ~ a m m a x b ~b (2 - k ) ~ .- - - - O < ~ t < r t n ~< k< nm+ l JS i n ce b k r - k ) = 1 f o r k = 0 , 1 , 2 , . . . , w e o b t a i n

I 1 1 1 _ < c am .O < m < r ( 1 0 )~ o ~ , let us e s t im a t e I1~'11~ from below. B y ( 7 ) , f o r a n y m = 0 , 1 , . . . , r w e h a v e

1 1 "~ 1 n~ +l -- 1 1 ~ 1v ~ ( t ) w , , ~ ( t ) d t > v ~ ( t ) ) " d t = v ~ ( t ) d t = b ~ 2 - ~ - 1 = -2 k +-----2 - - '4

]~: nm k--~- nm

H e n c e ( 8 ) i m p l i e s t h a t I I v : < l l F _ > 1 / 4 - T h e r e f o r e , b y ( 10 ) ,1 I l v l l EI 1 1 1 > - { a . < l l v l l } > _ 4 am _> i 4 c

T h i s t o g e t h e r w i t h ( 9) m e a n s t h a t t h e n o r m s o f t h e sp a c e s E a n d F a r e e q u i v a l e n t o n t h e l in e a r h u l l o ft h e s e t o f f u n c t i o n s v m ( m = 0 , 1 , . . . ) . H e n c e th e r e e x is ts a B > 0 s u c h t h a t , f o r a r b i t r a r y a , ~ ,

B -1


8/10

w S u f f i c i e n c y o f c o n d i t i o n (B ) f o r t h e o p e r a t o r I : E -+ F t o h a v e t h e D S S p r o p e r t yT h e o r e m 4 . Suppos e t h a t f un c t i on s ~ E G and r E G s a t i s fy cond i t ion ( A ) , ~ , < 1 , and w e hav eM (~ ) C A(~ b ) . T he n t he ope r a t o r I : M (~ ) --+ A(~b ) has t he D S S proper ty .

F i r s t , w e p r o v e t h e f o l l o w i n g a u x i l i a r y a s s e r t i o n .L e m m a . Und e r t he a s s um p t i o ns o f T he or e m 4 , the r e e z i s ts a f unc t i on p E G s uc h t ha t

( 1 ) l i m p( t )~-~o ~ = 0 ;(2 ) M ( ~ C A ( r

P r o o f . S i n c e 8~ , < 1 , [ 8, p . 1 5 6] i m p l i e s t h a tI I~ II M < ~> ~ s u p { ~ , ( t ) a : * ( t ) } .

o< t_< l

T h e r e f o r e , t h e re l a t i o n M ( ~ ) C A ( r i s e q u i va l e nt t o

f 0 d r < o o .~ ( s ) ( 1 1 )S i n c e t h e f u n c t i o n ~o i s c o n c a v e , w e h a v ef 0 d e ( s ) ~ r - t ' ) - r - k - l )~ ~ = 0 ~ , ( 2 - ~ )

h e n c e (1 1 ) i s e q u i v a l e n t t o t h e c o n d i t i o n~

r - k ) - r - k - x )~ ( 2 - ~ )k= O

< o o . ( 1 2 )

P u ta k = r - ~ ) - r - k - ~ ) a n d

T h e n S ,~ --+ 0 , a n d [ 1 2 , C h a p . 3 , E x . 1 2] a n d ( 1 2 ) i m p l y t h a tSn = ~ akk = n ~ ( 2 - k ) "

k =0 v / ~ - ~ ( 2 - k ) < o o . ( 13 )B y t h e d e f i n i t i o n o f u p p e r d i l a t i o n , t h e r e e x is t u > 0 a n d C > 0 s u c h t h a t g ~ + u < 1 a n d

M ~ ( t )


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H e n c e , b y ( 1 4 ) ,

s u p ~ ( 2 i t ) v / g ( 2 i t ) < C2~ '/22 (~ '+~ ' )/ for j = 0 , 1 , . . . .0 0 a n d C > 0 s u c h t h a t

r < C t " ( 1 7 )r -

w h e n e v e r 0 < t < 1 a n d 0 < a < 1 . T h i s a n d t h e c o n c a v i t y o f r g i v e

S. V. Astashkin- Disjointly Strictly Singular Inclusions of Symmetric Spaces

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Transcript of S. V. Astashkin- Disjointly Strictly Singular Inclusions of Symmetric Spaces