12 CALDERON'S FORMULA AND A DECOMPOSITION OF L 2 (Rn) The proof of this theorem is based on the following lemm a tha t tells us that m Q an d m p ar e "almost orthogonal": LEMMA (1.15) . If m p and m Q are (a , e) smooth molecules for dyadic cubes P and Q, where l(P) 1{Q) and a e, then / m p (x)mQ(x)dx c l(P) U(Q)\ a+n/2 1 + \XQ-XP\ KQ) -(n+e) We will prove this result in the Appendix, part I. PROOF O F THEORE M (1.14) . W e assume (1.15 ) and , thus, provisionally , that a e. I t clearly suffices to establish the desired inequality for a finitely nonzero sequence {s Q } (wit h c(n, a, e) independent of the sequence). The n f(J2spmp)(HsQmQ)^2 E \ S PWSQ\\[ s * E ww[m] 1{P)KQ) a+n/2 mpmQ l(P)KQ) = 2 £ {\s P \ l(P)l(Q) \XQ X p \ /(/) KQ) \s0\ i(py\ KQ) (a+n)/2 a/2 1 + 1 + \XQ X p KQ) ^-(n+£)/2 , -(n+e) KQ) \XQ-Xp\ KQ) -(«+£)/2 2AX/2Bl/2, by the Cauchy-Schwarz inequality, where *-T. E w 1 E E 2 (v-M)a 1 + \XQ - X p | /(G) -(n+e) and G ^= 0 {/ : /(/ )=2_"/(e)} But, we claim, 1 + l*Q ~ *p\ KQ) 1 ~(n+e) E KQ)=2- 1 + /(G) c^(l + \k\r {n+E) =c(n,e). kez" To se e this inequalit y w e first observe tha t th e first sum is periodi c wit h period 2~ u k, wher e k i s any of the canonical basis vectors (1 , 0, ... , 0), (0, 1 , ... , 0), ... , (0, 0, ... , 1). Thus , without los s of generality, we can
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