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