1. Calderon' s Formula and a Decomposition of L 2 (R") Let cp\ R n - C and , a s before , pu t q t (x) = C n q{Cxx) fo r t 0 . Observe that if tp(x) = cn{\ + |x|2)~(/l+1)/2 the n p , i s the Poisson kernel P t . In th e Introductio n w e presente d th e decompositio n (0.12 ) obtaine d fro m formula (0.10 ) i n which we used th e kernel t(dPJdt) w e also said tha t i n this section we shall present a variant o f thi s formula base d o n a compactly supported kernel . Th e following lemm a is needed for obtainin g this variant. LEMMA (1.1). Fix N e Z + . Then there exists a function (p\ R n -* R such that (1) Supp^c{xeR r t : |JC | 1 ) 5 ^ ( 0 ) (2) (p is radial (3) peC°°(R n ) (4) f r xy(p(x)dx = 0 if\y\N, yeZl, x y = xlx y 2 -xl», |y | = (5) f 0 °°mZ)]2dt/t=l ifieR"\{0}. REMARKS. Th e fac t tha t (p is real-value d an d radia l implie s tha t (p is real-valued thus , [p(t£)] 2 = \p(t£)\ 2 . Also observe tha t (1 ) an d (3 ) tel l u s that tp e 3f(R n ). PROOF. Le t 0 : R n - R b e a nonzero, radia l C°°-functio n wit h suppor t in B { (0). Pu t h =A k 6 fo r som e integer k N/2 (wher e A i s the Laplace operator). I t i s the n obviou s tha t h satisfie s (1) , (2 ) an d (3) . Integratio n by parts give s us (4) . Sinc e we have no t normalize d h , ther e i s n o reaso n to expec t (5 ) t o b e satisfied . W e claim , however , tha t f£°[h(t£)] 2 dt/t i s an absolutel y convergen t integral . T o se e this w e first observe tha t th e fac t that h 3 implie s tha t h(t£) i s rapidl y decreasin g a s t - o o fo r eac h £ ^ 0 . Thus , ff°[h(t£)] 2 dt/t i s absolutely convergent. Sinc e h satisfie s (4) , h(0) = 0 thus , h(t{) = O(t) a s t - 0+, an d thi s las t property guarantee s that /JlA^Of y i s absolutely convergent . Finally, since h i s radial so is h but this implies that c = / 0 °°[^(^)]2^ depends only on |£| , for all { ^ 0 . Furthermore , this integral is also indepen- dent o f the value of | sinc e y i s the Haar measur e fo r th e multiplicativ e 7 http://dx.doi.org/10.1090/cbms/079/02
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