16 CALDERON' S FORMULA AND A DECOMPOSITION OF L 2(R")
On the other hand, suppose \x-y\ l(Q) an d x e l\fnQ . I n this case y
certainly belongs to \Qy/nQ and , if (x - z ) o r (y - z) e 3( 2 w e must have
\z\ Al{Q) wit h A = A(n). Thus , using the cancellation property (1.18) we
have
(TaQ)(x) - (Ta Q)(y) = J K(z)[a Q(x - z) - a Q(y - z)]dz = 1 + 11 + 111,
where
z\3\x-y\
and
"-i
-I
•/3|jc--||r|:
K{z)[aQ{x - z) - a Q{x)]dz,
|*|31JC—-|
K(z)[aQ(y-z)-aQ(y)]dz
III = I K(z)[a Q(x - z) - aJy - z) ] dz.
z\Al(Q) *
U
Using (1.8) (wit h \y\ = 1) an d (1.16), we easily se e that th e absolute value s
of / an d 7 7 d o not excee d
r3|*-y|
d(Q)
-(l+»/2) f dr = 3cl(Q)-
(i+n/2)\y-x\
Jo
\x-y\r
C\Q\
-1/2
L KQ)
since a G (0, 1) an d \x - y\ l(Q) .
Finally,
rAI(Q)
\III\cf r~
nrn-l\x-y\l(Q)-{n/Ul)dr
J3\x-y\
Al{Q)
= c\x-y\\Q\
-1/2
/(G)"1
log
3 | * - y |
C\Q\
-1/2
\*-y\
KQ)
since a 1, and we obtain the desired conclusion about the action of T o n
the functions a Q.
Historical notes and comments. Th e basic idea in the Calderon formula o f
writing / a s a su m o f convolution s K
v
* / goe s back a lon g way . A ver-
sion of this identity appears in §34 of Calderon's 1964 paper [9 ] on complex
interpolation (cf . als o Hormander [45 ] and Peetr e [64]) . Althoug h the local-
ization ste p ha s antecedent s (fo r example , th e Shanno n samplin g theorem ,
which shal l b e discusse d i n §6) , i t i s curious tha t th e Caldero n identity , a s
we have developed i t here , was not systematicall y exploite d sooner . A sim-
ilar formula , somewha t mor e lik e th e on e i n Theore m (1.4), appears i n th e
1975 pape r [13] b y Caldero n an d Torchinsk y an d th e 1977 articl e [12] by
Calderon, both in the setting of parabolic Hardy spaces. Anothe r variant was
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