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~ n rn-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 196 4 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 197 7 articl e [12 ] by Calderon, both in the setting of parabolic Hardy spaces. Anothe r variant was
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