8 CALDER6N-S FORMULA AND A DECOMPOSITION OF L

2(R")

„ - l ]

group of positive rea l numbers . Lettin g p = c / / w e obtai n th e desire d

function, a

THEOREM

(1.2) (Calderon's Formula). Suppose peL

l(Rn)

is real valued,

radial and satisfies condition (5 ) in (1.1). Then, if f £ L

(1.3) / W = ^°°(^*^*/)Wy .

').

REMARK.

Thi s last equality is to be interpreted in the following L sense :

ifOeJoo an d

f6

dt

then \\f - f

E)s\\L2VBLn)

-• 0 as £ - 0 and S -•cx.

PROOF.

O n a formal leve l (1.3) is an immediate consequence o f equality

(5): th e Fourier transform of the right side of (1.3) is /({) J

0°°[p(^)]2f

=

/(£) • 1. The proof consists of a justification o f this argument. Suppose , for

Then, by Fubini's theorem,

6 A* r&

the moment, that / e

L1

n

L2

JR" Je l Je

[m)fdi.

Since ||p

(

* pt * /||2 \\p\\\\\f\\2 oo, we have ||/ £(5||2 // ||p||*IL/ll2T =

||p||J||/||2log(f). Hence , using Plancherel's theorem,

J^Jf~f^h-^eJ^Jf-Lsh

= c

m

lim

+0+,J-oo

/J/o{i-/W*}

d£,

1/2

cSr.-.,

But, fro m (5) , w e hav e |/(£){ 1 - /;[p(tf)]

2f

} | |/«) | an d a n applica -

tion of the Lebesgue dominated convergence theorem gives us lim £_0+ ^ ^

||/ - f

e

s\\2 = 0. Hence , (1.3), in the sense we described, is true for / e

L'nV. I f we only suppose / e L w e choose a sequence {fj} o f function s

in L f l i convergin g to / i n the L nor m and we leave th e rest of this

easy exercise to the reader. D

We can no w establis h th e versio n o f (0.12) base d o n th e kerne l g t; i n

terms tha t wil l b e define d a little later , thi s is an exampl e o f an "atomic"

decomposition of L

2(Rn).

W e use the notation involvin g cube s Q, multi-

defined i n the Introductio n an d in indices y eZ" an d thei r "norms"

Lemma (1.1

E

Q

fo r ECGf •

Lemm a (1.1) ; moreover , w e let

Dy

= (d

yi/dxyll)

•

-(dy"/dxynn)

an d writ e