8 CALDER6N-S FORMULA AND A DECOMPOSITION OF L 2 (R") - l ] group of positive rea l numbers . Lettin g p = c / / w e obtai n th e desire d function, a THEOREM (1.2) (Calderon's Formula). Suppose peL l (Rn) is real valued, radial and satisfies condition (5 ) in (1.1). Then, if f £ L (1.3) / W = ^°°(^*^*/)Wy . '). REMARK. Thi s last equality is to be interpreted in the following L sense : ifOeJoo an d f6 dt then \\f - f E) s\\L2VBLn) -• 0 as £ - 0 and S -•cx. PROOF. O n a formal leve l (1.3 ) is an immediate consequence o f equality (5): th e Fourier transform of the right side of (1.3) is /({) J 0 °°[p(^)]2f = /(£) 1. The proof consists of a justification o f this argument. Suppose , for Then, by Fubini's theorem, 6 A* r& the moment, that / e L1 n L2 JR" Je l Je [m)fdi. Since ||p ( * pt * /||2 \\p\\\\\f\\2 oo, we have ||/ £(5 ||2 // ||p||*IL/ll2T = ||p||J||/||2log(f). Hence , using Plancherel's theorem, J^Jf~f^h-^ e J^Jf-Lsh = c m lim +0+,J-oo /J/o{i-/W*} d£, 1/2 cSr.-., But, fro m (5) , w e hav e |/(£){ 1 - / [p(tf)] 2 f } | |/«) | an d a n applica - tion of the Lebesgue dominated convergence theorem gives us lim £ _0+ ^ ^ ||/ - f e s \\2 = 0. Hence , (1.3) , in the sense we described, is true for / e L'nV. I f we only suppose / e L w e choose a sequence {fj} o f function s in L f l i convergin g to / i n the L nor m and we leave th e rest of this easy exercise to the reader. D We can no w establis h th e versio n o f (0.12) base d o n th e kerne l g t i n terms tha t wil l b e define d a little later , thi s is an exampl e o f an "atomic" decomposition of L 2 (Rn). W e use the notation involvin g cube s Q, multi- defined i n the Introductio n an d in indices y eZ" an d thei r "norms" Lemma (1. 1 E Q fo r E CG f moreover , w e let Dy = (d yi /dxyll) -(dy"/dxynn) an d writ e
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