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?2 i
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 (1.11w ) 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
\x-y\7nl(Q)
\K(z)\l(Q)-{~{n/2)\z\dz
\z\7nl(Q)
lnl(Q)
Cl(Q)-
{~{n/2)
f r-
nrn-\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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