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