1. Calderon' s Formula and a Decomposition of L
2(R")
Let cp\ R
n
- C and , a s before , pu t q t(x) = C
nq{Cxx)
fo r t 0 .
Observe that if tp(x) = cn{\ +
|x|2)~(/l+1)/2
the n p , i s the Poisson kernel P t.
In th e Introductio n w e presente d th e decompositio n (0.12) obtaine d fro m
formula (0.10) i n which we used th e kernel t(dPJdt) ; w e also said tha t i n
this section we shall present a variant o f thi s formula base d o n a compactly
supported kernel . Th e following lemm a is needed for obtainin g this variant.
LEMMA
(1.1). Fix N e Z
+
. Then there exists a function (p\ R
n
-* R such
that
(1) Supp^c{xeR
r t
: |JC | 1 ) 5 ^ ( 0 ) ;
(2) (p is radial;
(3) peC°°(R
n);
(4) f
rxy(p(x)dx
= 0 if\y\N, yeZl, x
y
= xlx
y
2-xl», |y | =
(5) f
0°°mZ)]2dt/t=l
ifieR"\{0}.
REMARKS.
Th e fac t tha t (p is real-value d an d radia l implie s tha t (p is
real-valued; thus , [p(t£)]
2
= \p(t£)\
2.
Also observe tha t (1) an d (3 ) tel l u s
that tp e 3f(R
n).
PROOF.
Le t 0 : R
n
- R b e a nonzero, radia l C°°-functio n wit h suppor t
in B {(0). Pu t h =A
k6
fo r som e integer k N/2 (wher e A i s the Laplace
operator). I t i s the n obviou s tha t h satisfie s (1), (2 ) an d (3) . Integratio n
by parts give s us (4) . Sinc e we have no t normalize d h , ther e i s n o reaso n
to expec t (5 ) t o b e satisfied . W e claim , however , tha t f£°[h(t£)]
2
dt/t i s
an absolutel y convergen t integral . T o se e this w e first observe tha t th e fac t
that h 3 implie s tha t h(t£) i s rapidl y decreasin g a s t - o o fo r eac h
£ ^ 0 . Thus , ff°[h(t£)] 2dt/t i s absolutely convergent. Sinc e h satisfie s (4) ,
h(0) = 0 ; thus , h(t{) = O(t) a s t - 0+, an d thi s las t property guarantee s
that /JlA^Of y i s absolutely convergent .
Finally, since h i s radial so is h ; but this implies that c = / 0°°[^(^)]2^
depends only on |£| , for all { ^ 0 . Furthermore , this integral is also indepen-
dent o f the value of | sinc e y i s the Haar measur e fo r th e multiplicativ e
7
http://dx.doi.org/10.1090/cbms/079/02
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