CALDERON'S FORMULA AND A DECOMPOSITION OF L 2 (Rn) 11 / i s given by (1.5), with £ Q \s Q \2 oo, then / e L 2 (Rn) an d \\f\\ 2 L 2{r) c2 Yin \SQ\2 On e c an P r o v e ^i s by some sort of Hilbert spac e argument, in which we treat the expansion as if it were an orthogonal one. W e shall carry out thi s argument i n a more general setting. Instea d o f atoms we deal with molecules, a class of functions tha t includes the atoms. A n important reaso n for doin g this i s that man y operator s in analysis map atoms int o molecule s (and not into atoms). Thi s allows us to obtain estimate s for these operator s by examining their behavior on atoms. The definition o f a molecule associated wit h a cube Q = {xeRn:a i x t b n i= 1,2,.. . , n} involves the "lower lef t corne r of Q ," namel y x Q = a = (a x , a 2 , .. . , a n ). We then have: DEFINITION. Le t a , e 0 . A function m Q i s called a n (a , e) smoot h molecule fo r a cub e Q c R n i f an d only i f i t satisfie s th e following thre e conditions: { \ x _ x \) ~ {n +e) (1.11) \m Q (x)\ |Gf1 / 2 1 1 + 4(G) f {0Txe R " K,w-« e o-)i ier"2{^?F M ™%{'-^T^} "+" (1.12) forx,yeR n (1.13) / m n {x)dx = 0. jRn U REMARKS, (i ) (1.11) tells us that m Q i s "localized" or "peaks" at Q thus, we can regard (1.11 ) to be a weak version o f (1.6). (ii) Instead o f the C°° smoothnes s assumed in (1.8) we require in (1.12) that m Q satisf y a Lipschitz conditio n o f order a . Moreover , thi s Lipschit z condition has a decay at oo . (iii) (1.13 ) is the minimal cancellatio n (i V = 0) i n (1.7). (iv) The functions a Q i n Theorem (1.4), up to a constant factor, are (a, e) molecules for any e 0 an d a e (0, 1]. THEOREM (1.14) . Suppose 0 a 1, 0 e and, for each dyadic cube Q, m Q is an (a, e) smooth molecule for Q. Let f = J2n s QmQ f or a scalar sequence {S Q }Q^ satisfying Y^Q\ S Q\" °°- Then the series defining f converges in L 2 (Rn) and \ I/ 2 l \2c(n,a,e) \J2\ S Q\ Q
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