CALDERON'S FORMULA AND A DECOMPOSITION OF L 2(R") 9
THEOREM
(1.4). Suppose f e L
2(Rn)
and N e Z
+
. Then there exist
coefficients {s Q}, 0 SQ oo, where Q ranges over the dyadic cubes, and
functions {a Q} in 2 =
&(Rn)
such that
0-5) / = E ¥ f i
;
Q
(1.6) Suppfl flc3G
(1.7) / x yaQ(x)dx = 0 if\y\N\
(1.8) su p \(D\)(x)\
cyl(Qrlyhn/2
for y e Z ^ ;
xeRn
(1.9) £ 4
= IU1|22-
Q
REMARKS.
Equalit y (1.5) i s to b e interprete d i n th e followin g sense : fo r
v e Z , fixed, th e functio n ]C/(Q)=2- "
S QUQ(X)
IS w e
^ define d fo r eac h x,
since, b y (1.6), th e su m involve s a t mos t a finite numbe r o f term s tha t ar e
nonzero. Furthermore , by (1.8) with |y | = 0 , thi s sum represents a bounded
function. Le t
j
//*) = E E
S QaQ(*)'
U=Z~J/(Q)=2~"
Then (1.5) mean s that lim
y-400
\\f - fj\\
2
= 0 .
The coefficients s
Q
an d th e "atoms " a
Q
depen d o n / ; th e constants c
y
in (1.8) depend o n n an d N, a s well as on y .
PROOF.
Choos e tp a s i n Lemm a (1.1) . B y Theore m (1.2) an d identit y
(1.3), we have
/ ( * ) = r I 9 t{x-y){9t*f)iy)dy^.
JO
JRn
*
For a dyadi c cub e Q c R
n,
le t r(g ) = G x [ ^ , l(Q)] . The n th e se t
{T(Q)}9 Q dyadic , is a collection of half cubes covering R"
+1=
{(x , t) : x e
M", / 0} whos e interiors ar e pairwise disjoint. Thus ,
'M = E / / «/( * - ^ * /)M ^T-
QJJT(Q) t
Let
1/2
I /"/ " 2 , tff i
Sr
*Q
-{IL^-'^^'T}
and, when s
Q
/ 0 , defin e (thes e are examples of "smooth atoms, " which we
define later )
(1.10) a Q(x) = ± [[ p t{x7y){pt*f){y)dy^.
SOJJT(Q)
'
l
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