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On the reproducing formula of Calderón. (English) Zbl 0886.42012

Summary: The classical Calderón reproducing formula reads \[ \lim_{y\to 0,z\to\infty}\int^z_y(\phi_t*\widetilde{\phi}_t* f)(x){dt\over t}=f(x). \] Here \(f\in L^2({\mathbf R}_n)\) and the limit is taken in the \(L^2\)-norm; \(\phi\in L^1({\mathbf R}^n)\) is a normalized function satisfying certain conditions, \(\widetilde\phi(x)=\phi(-x)\), \(\phi_t(x)=t^{-n}\phi(t^{-1}x)\). The purpose of the paper is to generalize this result to \(f\in L^p({\mathbf R}^n)\), Let \(h=\varphi*\widetilde\varphi\).
Theorem: Let \(1<p<\infty\) and \(h\) be a radial function on \({\mathbf R}^n\) such that \((a)\) \(h\in L^p({\mathbf R}^n)\) and \(h(x)\log|x|\in L^1({\mathbf R}^n)\), (b) \(\int hdx=0\), and (c) \(\int h(x)\log(1/|x|)dx=1\). Then, for each \(f\in L^{p'}({\mathbf R}^n)\), (i) \(F(x,y)=\int^\infty_y(h_t*f)(x)dt/t\) converges to \(f(x_0)\) nontangentially at \((x_0,0)\) for each \(p'\)-Lebesgue point \(x_0\) of \(f\); (ii) \(|F(\cdot,y)-f|_{p'}=O(1)\) as \(y\to 0\); (iii) \(|F(\cdot,y)|_\infty+|F(\cdot,y)|_{p'}=o(1)\) as \(y\to\infty\). The condition on \(h\) can also be formulated in terms of the Fourier transform of \(h\).
Results close to the above have been obtained independently by the reviewer in the case when \(h\) is replaced by a finite Borel measure \(\mathbf R^n\) (not necessarily radial).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C15 General harmonic expansions, frames
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