10 CALDERON' S FORMULA AND A DECOMPOSITION OF L 2 (R n ) Suppose x i 3Q an d (y, t) e T(Q)\ the n / l(Q) \x-y\. Con - sequently, ^— ^ 1 an d i t follow s fro m (1 ) i n (1.1 ) tha t tp t (x - y) = 0. Thi s gives u s property (1.6) . I t follow s tha t fo r an y fixed x th e su m S/(Q)=2"" s Qao(x} ^ as at m o s t a fi n *te numbe r of nonzero terms. Thi s shows that fj(x) i s a well-defined function . Fro m the definitions o f f. 9 s Qi a Q and T(Q) w e see that j "=-J/(G)=2_l/ = E £ f x f ft^-y){? t *f){y)dy^ i/=-y The last expression is precisely ^ ^ for e = 2~ 7_1 an d S = 2 J \ hence, (1.5) follows from Theorem (1.2). Using propert y (5 ) i n Lemma (1.1 ) an d Plancherel's theorem , w e obtai n (1.9): £4 = c » f°° /j^)lV(0|2^y = ll/ll2. Property (1.7) follows from (4 ) in (1.1) and Fubini's theorem. Finally, we need to show property (1.8 ) i s satisfied. Sinc e t e [^ , /((?) ] when (y,t)e T(Q) , w e have: \D\(x)\ 7 - / / \[D y x pt(x-y)](pt*f)(y)\dy^ S QJJT(Q) t S i/L^ ~ '"''' T f k \llrJ"- * /)WI ^T 'T(Q) l ) I 1JT(Q) \tr2M\(Drp) t (x-y)\2dy^\ 'T(Q) * J f8upi(D'f)(x)iVMy,"",",/V(e)i,/2 U€R 2 7 =2|yi+"+i/2[/(0]-w-"-,/2[/(G)r/2 f ^ V/ 2 IIDV H \— = Cy l {Q) -M-»'\ Theorem (1.4) has an easily proved converse. A version of this converse is: If atoms are defined as functions a Q havin g properties (1.6), (1.7), (1.8) and
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