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
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