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
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