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