CALDERON'S FORMULA AND A DECOMPOSITION OF L 2 (R") 13 assume x e Q u 0 = {x: 0 x i 2 u } an d th e abov e inequalit y follow s easily. Thus , /i€Z/(/)=2-^ i/=-o o P Modifying th e argumen t abov e t o take into accoun t th e fac t tha t there are 2*" dyadic cubes P c Q wit h l(P) = 2" / 7(Q), we obtain **Ei'flf£2 oo 2 v^ „-M«+* ) c(n, e)2 fin = c{n, e) J2 \s Q \2 53 2 m = c{n, e , a ) £ |s G |2. G 0= 0 e This gives us the desired estimate for ||/|| 2 . In order to finish the proo f o f Theore m (1.14) , w e hav e t o remov e th e hypothesis a e. Tha t this can be done is a consequence of the fact that, if a /} , then 5 time s a (ft, e) molecul e is an (a, e) molecule . T o see this fact all we need to show is that if (1.12) holds with power 0 (instea d of a) , then this inequality is true with power a i f we multiply the right side by 2. But, in case \x-y\ l(Q) , the n [|J C - y\/l(Q)] fi [\x - y\/l(Q)] a an d the desired conclusion is trivial. I n case \x - y\ 1{Q) then , using (1.11), \mQ{x) - m Q (y)\ \m Q {x)\ + \m Q {y)\ •1/2 \Q\' + 1 + \y-xQ\ l(Q) 2\Qfl/2 su p M\x-y\ 1 + \x-z- x Q \ KQ) 2|G I -1/2 \HQ)) M sup 1 + \X-Z-XQ\ KQ) D We end this section by showing that a Calderon-Zygmund singular integral operator maps atoms into molecules. I n order to avoid irrelevant technicali- ties we consider a rather special class of such operators. Th e ideas presented here, however, can be used to study the most general Calderon-Zygmund op- erators on the large class of spaces that will be presented in this monograph (see Theorem (8.13) in Chapter 8). More precisely, we consider a convolution operator T o f the form (Tf){x) = j K{x-y)f{y)d yi
Previous Page Next Page