10 CALDERON' S FORMULA AND A DECOMPOSITION OF L

2 (Rn)

Suppose x i 3Q an d (y, t) e T(Q)\ the n / l(Q) \x-y\. Con -

sequently, ^— ^ 1 an d i t follow s fro m (1) i n (1.1 ) tha t tp t(x - y) =

0. Thi s gives u s property (1.6). I t follow s tha t fo r an y fixed x th e su m

S/(Q)=2""

sQao(x}

^

as at m o s t a

fi

n*te

numbe r of nonzero terms. Thi s shows

that fj(x) i s a well-defined function . Fro m the definitions o f f.

9

s

Qi

a

Q

and T(Q) w e see that

j

"=-J/(G)=2_l/

= E £ f

x

f ft^-y){? t*f){y)dy^

i/=-y

The last expression is precisely ^ ^ for e = 2~

7_1

an d S = 2

J

\ hence, (1.5)

follows from Theorem (1.2).

Using propert y (5 ) i n Lemma (1.1 ) an d Plancherel's theorem , w e obtai n

(1.9):

£4

= c»

f°°

/j^)lV(0|2

^y = ll/ll2.

Property (1.7) follows from (4 ) in (1.1) and Fubini's theorem.

Finally, we need to show property (1.8) i s satisfied. Sinc e t e [^ , /((?) ]

when (y,t)e T(Q) , w e have:

\D\(x)\ 7 - / / \[D

y

xpt(x-y)](pt*f)(y)\dy^

SQJJT(Q)

t

S

i/L^ ~ '"''' T f k \llrJ"- *

/)WI

^T

'T(Q) l ) I 1JT(Q)

\tr2M\(Drp)t(x-y)\2dy^\

'T(Q) * J

f8upi(D'f)(x)iVMy,"",",/V(e)i,/272RU€

=2|yi+"+i/2[/(0]-w-"-,/2[/(G)r/2

f ^

V/ 2

IIDV H

\—

= Cyl{Q)-M-»'\ •

Theorem (1.4) has an easily proved converse. A version of this converse is:

If atoms are defined as functions a

Q

havin g properties (1.6), (1.7), (1.8) and