CALDERON'S FORMULA AND A DECOMPOSITION OF L
2(Rn)
11
/ i s given by (1.5), with £
Q
\s
Q\2
oo, then / e L
2(Rn)
an d \\f\\
2
L2{r)
c2
Yin
\SQ\2

On e c an
P
r o v e
^i s by some sort of Hilbert spac e argument, in
which we treat the expansion as if it were an orthogonal one. W e shall carry
out thi s argument i n a more general setting. Instea d o f atoms we deal with
molecules, a class of functions tha t includes the atoms. A n important reaso n
for doin g this i s that man y operator s in analysis map atoms int o molecule s
(and not into atoms). Thi s allows us to obtain estimate s for these operator s
by examining their behavior on atoms.
The definition o f a molecule associated wit h a cube
Q =
{xeRn:ai
x
t
b
n
i= 1,2,... , n}
involves the "lower lef t corne r of Q ," namel y x
Q
= a = (a x, a 2, .. . , a n).
We then have:
DEFINITION.
Le t a , e 0 . A function m
Q
i s called a n (a , e) smoot h
molecule fo r a cub e Q c R
n
i f an d only i f i t satisfie s th e following thre e
conditions:
{ \
x
_
x
\) ~
{n+e)
(1.11) \m Q(x)\ |Gf1/2 1 1 + 4(G) f {0Txe R ";
K,w-«eo-)i
ier"2
{^?F
M
™%{'-^T^}
"+"
(1.12) forx,yeR
n;
(1.13) / m n{x)dx = 0.
jRn U
REMARKS,
(i ) (1.11) tells us that m
Q
i s "localized" or "peaks" at Q ; thus,
we can regard (1.11 ) to be a weak version o f (1.6).
(ii) Instead o f the C°° smoothnes s assumed in (1.8) we require in (1.12)
that m
Q
satisf y a Lipschitz conditio n o f order a . Moreover , thi s Lipschit z
condition has a decay at oo .
(iii) (1.13) is the minimal cancellatio n (i V = 0) i n (1.7).
(iv) The functions a
Q
i n Theorem (1.4), up to a constant factor, are (a, e)
molecules for any e 0 an d a e (0, 1].
THEOREM
(1.14) . Suppose 0 a 1, 0 e and, for each dyadic cube
Q, m
Q
is an (a, e) smooth molecule for Q. Let f = J2n
sQmQ
f
or a
scalar sequence
{S Q}Q^
satisfying
Y^Q\
SQ\"
°°- Then the series defining
f converges in L
2(Rn)
and
\
I/ 2
l\2c(n,a,e)
\J2\
SQ\
Q
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