By Miguel de Guzmán (Eds.)
ISBN-10: 0444861246
ISBN-13: 9780444861245
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Extra resources for Real Variable Methods in Fourier Analysis
Example text
The Kolmogorov con- d i t i o n i n S e c t i o n 3 c o n s t i t u t e s a n o t t o o wellknown b u t v e r y n i c e t o o l t o s t u d y t h e t y p e o f an o p e r a t o r . The common f e a t u r e i n t h e techniques o f i n t e r p o l a t i o n and e x t r a polation i s the following. Assume t h a t we know t h a t an o p e r a t o r T behaves w e l l on some spaces o f a c e r t a i n f a m i l y o f f u n c t i o n spaces. Can one say a n y t h i n g about i t s behaviour on t h e i n t e r m e d i a t e spaces o f t h a t f a m i l y ( i n t e r p o l a t i o n ) o r on t h e extreme cases o f t h a t family ( e x t r a p o l a t i on) ?
For x E Ah containing x such t h a t x1 11 f lQ(x)l < where Qt x . Then, 6 a h each have,i6 t h e r e i s a l a r g e s t dyadic i n t e r v a l If1 > A Clearly Q(x) . hi 111 X > O , we Q(x) and so i t i s obvious t h a t ( Q ( X ) ) ~ s~ a~t i s f y t h e f i n i t e ascending chain condition. 4. obtaining Q, disjoint such t h a t Ah = U Q, . Observe t h a t i s the f a t h e r of Q,. W i t h these i n e q u a l i t i e s the s t a t e m n t i s obvious. Aiyfication C d d m 6 n - Zyqmund decompob&on 2.
S P1 such t h a t , ... , S pi 6 21 , such t h a t p2 Clearly we have f o r each n This proves t h e lemma. Phaad ad ,the. Theaheni 2 . 2 . 1 . Assume T* i s not of weak type (p,p). Then, i f we f i x a sequence c k 4 00 , c k > 0 , t h e r e e x i s t s a sequence { f k I C L p , f k a 0 and a sequence X k > 0 such that Let us c a l l gk = fk , Ak = {T*gk > 1 I. We can w r i t e , 2. FINITENESS AND THE TYPE 22 Let hk hk copies Ai , A; , ... , A;k u(Ak) c 1 6 hk be a n a t u r a l number such t h a t 2.