S.V. Astashkin- Interpolation of Operators in Quasinormed Groups of Measurable Functions

8/3/2019 S.V. Astashkin- Interpolation of Operators in Quasinormed Groups of Measurable Functions

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35, No. 6, 1994

O F O P E R A T O R S I N Q U A S I N O R M E DG R O U P S O F M E A S U R A B L E F U N C T I O N S

s . v . A s t a s h k i n U D C 5 17 .9 82 .2 7e c a l l t h a t a q u a s i n o r m o n a n A b e l i a n g r o u p X i s a r e a l f u n c t i o n [ [ - [ I x t h a t i s d e f i n e d on X a n d

a t i s f ie s th e co nd i t io ns : ( a ) [ [x [ lx _> 0 , [ [x l [ = 0 -'. ;- x = 0 ; (b ) l [ - x l l x = [ [x [ [x ; an d (c ) [ Ix + 1 . In th i s ca se X i s ca l le d a q u a s i n o r m e d g r o u p . A s i s k n o w n ,u t l os s o f g e n e r a l i t y w e m a y a s s u m e t h a t C = 1 [1 , p p . 8 0 - 81 ] . I t th e p r e s e n t a r t i c l e w e s t u d yc o m p l e t e q u a s i n o r m e d g r o u p s X o f L e b e s g u e m e a s u r a b l e r e a l f u n c t i o n s o n t h e s e m i a x i s ( 0, c ~ ) w i t hh e c o n v e n t i o n a l a d d i t i o n . W e a s s u m e t h a t t h e q u a s i n o r m [ [- ][ x s a ti s fi e s t h e f o ll o w i n g t w o c o n d i t io n s :( 1 ) i f I x[ < [ y[ ( i . e . , I x ( s ) [ < [ y ( s ) [ a . e . ) a n d y E X t h e n x E X a n d [ I x [ Ix _< [ [ Y [ [ x ; ( 2 ) i f f u n c t i o n sIx [ a n d [y [ a r e e q u i m e a s u r a b l e ( i .e . , n l z l ( r ) = n l y l ( r ) , n z ( r ) = p { s > 0 : z ( s ) > r } ) a n d y E X t h e nx a n d I I l lx = I l Y l I x . S u c h a q u a s i n o r m e d g r o u p w i l l b e c a l l e d a r e a r r a n g e m e n t i n v a r i a n t g r o u p( r .i . g r o u p ) . A n i m p o r t a n t e x a m p l e o f a n r .i . g r o u p i s t h e s c a l e L p = L p ( 0 , c ~ ) , 0 < p < c ~ , d e fi n e d ina r o u t i n e f a s h i o n . F o r 0 < p < 1 , t h e q u a s i n o r m i n L p i s d e f i n e d b y t h e e q u a l i t y

~0~l x l l p = I x ( s ) l p d s .T h u s , f o l l o w i n g [ 2] w e c a n p a s s t o t h e l i m i t c a se : p = 0 . W e d e n o t e b y L 0 t h e s e t o f a l l m e a s u r a b l ef u n c t i o n s x = x ( t ) f o r w h i c h ] lx ll 0 = # ( s u p p x ) < o o , w h e r e s u p p x = { t > 0 : x ( t ) ~ 0 } . T h e n L 0b e c o m e s a n r . i. g r o u p c o n t i n u o u s l y e m b e d d e d i n t o t h e s p a c e S o f a ll m e a s u r a b l e a . e. f i n it e fu n c t i o n se n d o w e d , a s u s u a l , w i t h c o n v e r g e n c e i n m e a s u r e o n t h e s e ts o f f i n it e m e a s u r e . I n th i s c o n n e c t i o n w em a y c o n s i d e r t h e B a n a c h p a i r ( L 0 , L o ~) of r . i. g r o u p s w h o s e r o l e i n t h e c l a s s o f a l l r . i. g r o u p s i s s i m i l a rt o t h e r o l e o f t h e p a i r ( L 1 , L o o) i n t h e c l a ss o f 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 s o n ( 0 , o c ) [ 3 ].T h e m a i n g o a l o f t h e a r t i c l e i s t o d e s c r i b e t h e o r b i t O r b ( a ; L 0 , L o o) i n t h e p a i r ( L 0 , L o o) f o r a na r b i t r a r y f u n c t i o n a E L 0 + L o~ . W e h a v e as a c o r o ll a r y s o m e t h e o r e m c l a i m i n g t h a t a ll r .i . g r o u p si n t e r m e d i a t e b e t w e e n L 0 a n d L oo a r e i n t e r p o l a t e w i t h re s p e c t t o t h e p a ir . M o r e o v e r , it i s s h o w n t h a t ,g e n e r a l l y s p e a k i n g , t h e E - o r b i t o f a n e l e m e n t a E L 0 + L oo d o e s n o t c o i n c i d e w i t h i t s o r b i t . T h e s er e s u l ts t e s t i f y t o e s s e n t i a l d i s t i n c t io n b e t w e e n i n t e r p o l a t i o n i n t h e p a i r ( L 0 , L oo ) a n d t h a t i n t h e p a i r( L 1 , L o o ) [3 , C h a p t e r 2 , w1 . D e s c r i p t i o n f o r O r b ( a ; L 0 , L o o ) . A m a p p i n g T : X ~ X , w h e r e X is a n r .i . g r o u p , is c a ll eda h o m o m o r p h i s m i f T ( x + y ) = T x + T y a n d T ( - z ) = - T ( x ) fo r z , y e X . A s u s u a l , a h o m o m o r p h i s mis ca l led b o u n d e d i f

[ [ T I [ x ~ x = s u p ~ < o c .I l x l lL e t ( X 0 , X 1 ) b e a B a n a c h p a i r o f r .i . g r o u p s . W e d e fi n e t h e o rb i t o f a n e l e m e n t a G X 0 + X 1 t o b et h e s e t o f a ll x E ) t o + X 1 r e p r e s e n t a b l e a s z = T a , w h e r e T i s a b o u n d e d h o m o m o r p h i s m i n X 0 a n dX 1 . F u r t h e r m o r e ,

l[xl]Orb = in f l l T l l ( x o , x , ) ,w h e r e t h e i n f i m u m i s t a k e n o v e r a ll h o m o m o r p h i s m s T w i th T a = x; [[T]](Xo ,X1) = m ~,~ HT [[x , . . . x , .

I t is e a s y t o c h e c k t h a t t h e q u a s i n o r m H " [[o rb m a k e s O r b ( a ; X 0 , X I ) i n t o a n r .i . g ro u p .H e n c e f o r t h w e s h al l d e n o te b y z * ( t ) t h e n o n in c r e as i n g r e a r r a n g e m e n t o f a m e a s u r a b l e f u n c t i o n[z(t)[ [3, p . 83].Sam ara . Trans la ted f rom Sibirski~ M atematicheski~ Zhurnal, Vol. 35, No. 6, pp. 1215-1222, November-December,

1994. Original article submitted November 25, 1993.0037-4466/94/3506--1075 $12.50 (s 1994 Plenum Publishing Co rporatio n 1075


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T h e o r e m 1 . G i v e n a n a r b i t r a r y a 9 Lo + Loo, the orb i t O r b ( a ; L o , L o o ) c o in c id e s w i th th e se to f a / / x 9 L 0 + L o o f o r w h i c h t h e r e e x i s t s C > 0 s u c h t h a t

, * ( t ) < C a * ( t / C ) . (1 )M o r e o v e r , [ [ z [ l o rb e q u a l s t h e g r e a t e s t l o w e r b o u n d o f a l l C ' s f o r w h i c h i n e q u a l i t y ( 1 ) i s sa t i s f i e d .

F i r s t w e e s t a b l i s h o n e a u x i l i a r y c l a i m . G i v e n a p a i r ( X 0 , X 1 ) o f r .i . g r o u p s , w e i n t r o d u c e t h ef u n c t i o n a l

E ( t , z ; X o , X 1 ) = i n f { l l x - ~ 0 1 1 x , ; I I ~ 0 1 1 x 0 - -- t } , ~ 9 N o + x l , t > 0 ,t h a t p l a y s a n i m p o r t a n t r o l e i n a p p r o x i m a t i o n t h e o r y [1 , C h a p t e r 7 ].

L e m m a 1 . I f T i s a b o u n d e d h o m o m o r p h i s m i n X o a n d X 1 t h e n / o r z 9 X o + X 1 t h e f o l lo w i n gi n e q u a l i t y h o l d s :E ( t , T x ; X o , X 1 ) _ < I I T I I E ( t / I I T I I , ~ ; X o , X ~ ) ,

w h e r e I I T I I = I I T I l ( x o , X , ) .P R O O F . L e t y 0 = T z o a n d z 0 9 X 0 . I f I I T I I I I~ 0 1 1 x 0 _ t t h e n I l y 0 1 l x 0 _ < t . T h u s ,

E ( t , T x ; X o , X 1 ) = i n f { l J T z - y 0 1 l x , ; I lY 0 1 1 x 0 - < t }< i n f { l l T z - T ~ 0 1 1 X l ; I I~ l l x 0 - < t / l lTI I} l } , E n = t : < ] a ( t ) l < h E N .n + l - - '

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I f z* ( o o ) = 0 t hen T4 i s t he s am e a s i n t he p r ev i ou s ca s e . I f z* ( c~ ) = 61 > 0 t he n , f o r x l ( t ) =m a x ( l z ( t ) l , $ a ) , find a m e a s u r e - p r e s e r v i n g m a p p i n g w l : ( 0, c o ) ~ ( 0, o o ) s u c h t h a t- - 5 1 / 2 a n d , b y p u t t i n g z l = z x - z * (w a ) a n d a l ( t ) = z l ( t ) / x * ( w a ( t ) ) i f z* ( wa( t ) ) ~ 0an d ax ( t ) = 0 i f z* ( w l ( t ) ) = 0 , ob t a i n Ila all~o < ~ . C o n s e q u e n t l y , t h e h o m o m o r p h i s m 7 "4 , T 4 y ( t) =(1 + a a ( t ) ) y ( w l ( t ) ) , i s b o u n d e d i n L 0 a n d L o o , I ITaII(Lo,L~ ) -< 1 + e, T 4 x * - - x l . F i n a l l y , T s y ( t ) =f l t ( t ) y ( t ) , w h e r e f ~ l( t) " - - : g ( t ) / Xl ( t ) i f x l ( t ) r 0 a n d ~ l ( t ) = 0 i f Z l ( t ) - - 0 . S i n c e I f ~ l ( t ) l ~ .~ 1 , w e h a v eI ITs lI (L0,L.~) < 1 an d T s x l = x .B y p u t t i n g r = T s T 4 T a T 2 T 1 , w e o b t a i n IITII(Lo,L~) < (1 + ~)~ a n d T a = z a s i n c a s e ( a ) .

Now , l e t i ne qu a l i t y ( 1 ) be s a t is f i ed f o r s om e C > O. As s i gn y ( t ) = x ( t ) / C a n d b ( t ) = a ( t / C ) .T h e n y* (t) 0 t h e r e e x is t s a h o m o m o r p h i s m T w i t hI IT I I (Lo , L~ ) < - ( 1 + e ) 2 , T b = y .

D e fi ne t h e h o m o m o r p h i s m s R c z ( t ) = z ( t / C ) a n d Q c z ( t ) = C z ( t ) . Si nce I IRc l lLoo-~ /~ = I lQCllLo- .Lo- - X a n d I l Q c l l z | - I l R c l l Z o - . L o - C , for V -- Q c T R c w e h a v e [IVll(Lo,Loo) < (1 + e ) 2 C a n dVa~ -x .Si nce e > 0 is a r b i t r a r y , i t now f o l lows f r om ( 1 ) t ha t x E Or b ( a ; L0 , Loo) and I l z l lO~b ___ c .T o g e t h e r w i t h i n e q u a l i t y (3 ) i t m e a n s t h a t T h e o r e m 1 is p r o v en .2 . I n t e r p o l a t i o n i n t h e p a i r ( L 0 , L o o ) . R e c al l t h a t a n r .i . g r o u p X i s c a ll ed i n t e r p o l a t ebe t w een r . i . g r ou ps X0 an d X1 i f X0 n X1 C X C X o + X 1 ( w i t h a l l e m b e d d i n g s c o n t i n u o u s ) a n de v e r y h o m o m o r p h i s m b o u n d e d i n X 0 a n d X 1 is b o u n d e d i n X .W e e x h i b i t t w o a u x i l i a r y r e s u l t s .

L e m m a 2 . T h e f o l l o w i n g as s e r t i ons a r e v a l i d f o r e v e r y r . i . g r o u p X on (0 , oo) :( 1 ) i f x 9 x a n d C > 0 t h e n C x ~ X a n d I I C x l l x < - ( [C ] + a ) l l ~ l l x ( I V ] i s t h e i n t e g r a l p a r to ~ c ) ;(2) for each ~- > 0, the d i l a t a t i o n h o m o m o r p h i s m R ~ x ( t ) = x( t /~- ) i s b o u n d e d i n X a n d

I l e ~ l l x - ~ x


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O n . th e o t h e r h a n d , w e h a v eOo

H IE oI[ _ < ~ [ [ z i [ [ , n = 1 , 9 . , . . . ,t n

Irnns inc e l e o < ~ I zi ] by th e de f in i t ion o f E0 . Th us , [ l l E o l l = 0; i .e . , ~ ( E 0 ) = 0 . T h e c o n t r a d i c t i o n! 11o b t a i n e d p r ov e s t h e l e m m a .A s s u m e t h a t t h e q u a s i n o r m H" I Ix o n a n r .i . g r o u p X p o s se s se s t h e p r o p e r t y

i n f { [ [ x [ [ x ; z # 0 ) > 0 . ( 8 )T h e n i t g e n e r a t e s d i s c r e t e t o p o l o g y i n X . W e s h a l l c a l l a q u a. si n o rm w i t h p r o p e r t y ( 8) discrete.

T h e o r e m 2 . L e t X b e a n o n e m p t y r .i. g r o u p w i t h n o n d i sc r e te q u a si n o r m a n d l e t X # S , w h e r eS i s the se t o f a l l measurable a.e . f inite fun ctio ns on th e serniaxis. T h e n Lo [~ Loo C X C Lo + Loo;m o r e o v e r , t h e e m b e d d i n g s a r e cont inuous .P R O O F . A s s u m e t h a t z G LoNLoo; i . e ., a = # ( s u pp x ) < oo a nd ][x [[a o < oo (we ma y s uppos e th a ta > 0 ) . S i n c e X i s n o n e m p t y , t h e r e e x i st s a n z0 G X , x 0 # 0 . F u r t h e r m o r e , t h e r e e x i s t a m e a s u r a b l e

s e t E a nd a n e > 0 s uc h tha t 0 < # ( E ) = f~ < I I~ l iL o n L o o , ~ < I IX l iL o n L o o , an d z = eXE < [zo[ .Th ere by , z e X an d I [z l lx < [ [x01lx.D e f in e t h e f u n c t i o n w ( t ) = I l x l l o o a E ( ( M ~ ) t ) . B y L e m m a 2 , w G X a n dI l w l l x _< ( [ ~ / / ~ ] + 1 ) ( [ l l x l l o o / ~ ] + 1 ) l l z l l x 0 , f i n d x E X s u c h t h a t x # 0 a n d I l z l l x < 6 . A sa b o v e , t h e r e e x i st a 3' > 0 a n d a m e a s u r a b l e s e t F , # ( F ) > 0 , f o r w h i c h z = 7 1 F < I z l . T h u s ,I l z l l x < ~. I f n o w I I~1111L0~L~o - - ' 0 a n d I I~1111L0nLoo -----m i n ( 7 , # ( F ) ) f o r n > N , t h e n z~, < z *, w h e n c eI Iz1111x = I I * l l x < - I l z l l x < 6 . Consequently, I I~1111x - -* 0 .D e m o n s t r at e t h a t X 9~ Lo +L oo i f a n d o n l y i f X = S . I n d e ed , a s su m e t h a t t h e r e e x is t s a n x 0 E X ,z 0 ~ L 0 + L oo . T h e l a s t m e a n s t h a t t h e d i s t r i b u t i o n f u n c t i o n n l~ 01 (r ) i s i d e n t i c a l l y i n f i n i t e . S i n c eX i s a n r . i . g r ou p , a l l f unc t ion s z = z ( t ) f o r wh ic h n lz l ( r ) = c ~ , r > 0 , be lo ng to X . I f now x ( t ) isa r b i t r a ry t h e n I~ 1 _< z = m a x ( I z 0 1 , I~ 1) a n d % 1 ( ~ ) - ~ . T h u s , ~ e X .I n c o n c l u s i o n w e p r o v e t h a t t h e e m b e d d i n g X C L 0 + L oo i s c o n t i n u o u s . A s s u m e t o t h e c o n t r a r yt h a t t h e r e i s a s e q u e n c e {x11} C X s u c h t h a t I I ~ , l l x ~ 0 whe r e a s

I I~ 1 1 1 1 L 0 + L ~ > - ~ > 0 , n = 1 , 2 , . . . . (9 )F in d a su bse qu en ce { ~11} C {z11} for wh ich

OO

I * l lx 0 .

C o n s i d e r t h e f o l l o w i n g t h r e e c a s e s :1 . t


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R e f e r e n c e s1 . J . B e r g h a n d J . L 6 f s t r6 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 ia n t r a n s l a t i o n ] , M i r, M o sc o w(1980) .2 . J . P e e t r e a n d G . S p a r r , " I n t e r p o l a t i o n o f n o r m e d A b e l i a n g r o u p s , " A n n . M a t . P u r a A p p l . ( 4 ),92 , No . 4 , 217 - 262 ( 1972) .3 . S . G . Kr eYn , Yu . I . Pe t un i n , and E . M . SemSnov , I n t e r p o l a t i on o f L i nea r Op e r a t o r s [ in Rus s i an ] ,N a u k a , M o s c o w ( 1 9 7 8 ) .4 . V . I. O v c h i n n i k o v , " T h e m e t h o d o f o r b i t s i n i n t e r p o l a t i o n t h e o r y , " M a t h . R e p . , 1 , N o . 2, 3 4 9 - 5 1 5(1984) .5 . B . S . M i t y a g i n , " A n in t e r p o l a t i o n t h e o r e m f or m o d u l a r sp a c e s ," M a t . S b . , 6 6 , N o . 4 , 4 7 3 - 4 8 2(1965) .6 . A . - P . C a l d e r o n , " S p a c e s b e t w e e n L 1 a n d L ~ 1 76n d t h e t h e o r e m o f M a r c i n k ie w i c z ," S t u d i a M a t h . ,26 , 273 - 299 ( 1966) .

TRANSLATED BY V. N. DYATLOV

S.V. Astashkin- Interpolation of Operators in Quasinormed Groups of Measurable Functions

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Transcript of S.V. Astashkin- Interpolation of Operators in Quasinormed Groups of Measurable Functions