CALDERON'S FORMULA AND A DECOMPOSITION OF L 2 (R") 9 THEOREM (1.4) . Suppose f e L 2 (Rn) and N e Z + . Then there exist coefficients {s Q }, 0 SQ oo, where Q ranges over the dyadic cubes, and functions {a Q } in 2 = &(Rn) such that 0-5) / = E ¥ f i Q (1.6) Suppfl fl c3G (1.7) / x y aQ(x)dx = 0 if\y\N\ (1.8) su p \(D\)(x)\ cyl(Qrlyhn/2 for y e Z ^ xeRn (1.9) £ 4 = IU1| 2 2- Q REMARKS. Equalit y (1.5 ) i s to b e interprete d i n th e followin g sense : fo r v e Z , fixed, th e functio n ]C/(Q)=2- " S QUQ(X) IS w e ^ define d fo r eac h x, since, b y (1.6) , th e su m involve s a t mos t a finite numbe r o f term s tha t ar e nonzero. Furthermore , by (1.8) with |y | = 0 , thi s sum represents a bounded function. Le t j //*) = E E S QaQ(*)' U=Z ~J/(Q)=2~" Then (1.5 ) mean s that lim y-400 \\f - fj\\ 2 = 0 . The coefficients s Q an d th e "atoms " a Q depen d o n / th e constants c y in (1.8 ) depend o n n an d N, a s well as on y . PROOF. Choos e tp a s i n Lemm a (1.1) . B y Theore m (1.2 ) an d identit y (1.3), we have / ( * ) = r I 9 t {x-y){9t*f)iy)dy^. JO JRn * For a dyadi c cub e Q c R n , le t r(g ) = G x [ ^ , l(Q)] . The n th e se t {T(Q)}9 Q dyadic , is a collection of half cubes covering R" +1 = {(x , t) : x e M", / 0} whos e interiors ar e pairwise disjoint. Thus , 'M = E / / «/( * - ^ * /)M ^T- QJJT(Q) t Let 1/2 I /"/ " 2 , tff i Sr *Q -{IL^-'^^'T} and, when s Q / 0 , defin e (thes e are examples of "smooth atoms, " which we define later ) (1.10) a Q (x) = ± [[ p t {x7y){pt*f){y)dy^. S OJJT(Q) ' l
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