12
CALDERON'S FORMULA AND A DECOMPOSITION OF L 2(Rn)
The proof of this theorem is based on the following lemm a tha t tells us
that m
Q
an d m
p
ar e "almost orthogonal":
LEMMA
(1.15). If m
p
and m
Q
are (a , e) smooth molecules for dyadic
cubes P and Q, where l(P) 1{Q) and a e, then
/ m p(x)mQ(x)dx
c
l(P)
U(Q)\
a+n/2
1 +
\XQXP\
KQ)
(n+e)
We will prove this result in the Appendix, part I.
PROOF O F THEORE M
(1.14) . W e assume (1.15 ) and , thus, provisionally ,
that a e. I t clearly suffices to establish the desired inequality for a finitely
nonzero sequence {s Q} (wit h c(n, a, e) independent of the sequence). The n
f(J2spmp)(HsQmQ)^2 E
\ SPWSQ\\[
s * E ww[m]
1{P)KQ)
a+n/2
mpmQ
l(P)KQ)
= 2 £ {\s P\
l(P)l(Q)
\XQ
X
p
\
/(/)
KQ)
\s0\
i(py\
KQ)
(a+n)/2
a/2
1 +
1 +
\XQ
X
p
KQ)
^(n+£)/2
,
(n+e)
KQ)
\XQXp\
KQ)
(«+£)/2
2AX/2Bl/2,
by the CauchySchwarz inequality, where
*T.
E
w
1E
E 2
(vM)a
1 +
\XQ  X
p

/(G)
(n+e)
and
G ^= 0 {/ : /(/
)=2_"/(e)}
But, we claim,
1 +
l*Q ~ *p\
KQ)
1 ~(n+e)
E
KQ)=2
1 +
/(G)
c^(l + \k\r
{n+E)=c(n,e).
kez"
To se e this inequalit y w e first observe tha t th e first sum is periodi c wit h
period 2~
uk,
wher e k i s any of the canonical basis vectors (1, 0, ... , 0),
(0, 1, ... , 0), ... , (0, 0, ... , 1). Thus , without los s of generality, we can