14 CALDERCN' S FORMULA AND A DECOMPOSITION OF L 2 (Rn) where th e kerne l K i s supported outsid e a neighborhoo d {x: \x\ 3} o f the origin an d for a n a (0, 1) , satisfie s (1.16) \K(x)\C\x\- n \ (1.17) \K(x - 3 0 - K(x)\ C\y\a\x\—n whe n \y\ l -f and (1.18) f K{x)dx = 0, whenOi? 1 i? 2 00. Let a Q b e a functio n satisfyin g (1.6) , (1.7 ) wit h y = 0 , an d (1.8 ) wit h \y\ 1 . Le t u s sho w tha t Ta Q = Cm Q , wher e m Q i s a n (a , a ) smoot h molecule an d C i s independen t o f S an d th e functio n a Q satisfyin g th e above conditions . I t follow s fro m Theorem s (1.4 ) an d (1.14 ) tha t th e oper - ator nor m o f T o n L i s bounded independentl y o f S (thi s follow s b y a more direct argument if we use PlancherePs theorem however, the argument presented her e carrie s over , wit h modifications , t o th e nonconvolutio n cas e where PlancherePs theorem doe s not apply—see §8) . Th e fact tha t T map s an atom a Q int o a multiple of a molecule rrig shows that T behave s like a local operator . Although the proof that we now give appears technical, all of the estimates are elementary . Thi s illustrate s a basi c advantag e o f th e approac h w e ar e considering. We first observe that propert y (1.13 ) i s clearly satisfied (fro m eithe r (1.7 ) with y - 0 o r fro m (1.18)) . I n orde r t o obtai n (1.11 ) w e argue as follows : Suppose, first tha t x e lOy/nQ the n | x y| lnl{Q) whe n y e 3Q. Consequently, usin g (1.18), the mean value theorem, an d (1.8), \(TaQ)(x)\ = / K{x - y)[a Q (y) - a Q (x)] dy J\x-v\lnl(Q) * * J\z \K(z)\l(Q)-{~{n/2)\z\dz \z\7nl(Q) lnl(Q) Cl(Q)- { ~{n/2) f r- n rn-\dr = Cl(Q)' Jo in/2) If x i 1002( 2 an d y e 3Q , the n \y - x Q \ 2\/nl{Q) \\x-x Q \. Hence , by (1.17 ) an d (1.7) , (1.8) (wit h y = 0),
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