Boundedness of iterated spherical average

Abstract

The iterated spherical average \mathrm{\Delta }(A_1{)}^N is an important operator in harmonic analysis, and has very important applications in approximation theory and probability theory, where \mathrm{\Delta } is the Laplacian, A_1 is the unit spherical average and (A_1{)}^N is its iteration. In this paper, we mainly study the sufficient and necessary conditions for the boundedness of this operator in Besov-Lipschitz space, and prove the boundedness of the operator in Triebel-Lizorkin space. Moreover, we use above conclusions to improve the existing results of the boundedness of this operator in L^p space.

Keywords

Iterated spherical average / Besov-Lipschitz space / Triebel-Lizorkin space / $ L^{p} $ space

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