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

References

[1] Belinsky E, Dai F, Ditzian Z. Multivariate approximating average. J Approx Theory 2003; 125: 85–105

[2] Borwein J, Straub A, Vignat C. Densities of short uniform random walks in higher dimensions. J Math Anal Appl 2016; 437(1): 668–707

[3] Fan D, Lou Z, Wang Z. A note on iterated sperical average on Lebesgue spaces. Nonlinear Anal 2019; 180: 170–183

[4] Fan D, Zhao F. Approximation properties of combination of multivariate averages on Hardy spaces. J Approx Theory 2017; 223: 77–95

[5] GrafakosL. Modern Fourier Analysis. 2nd ed. Graduate Texts in Mathematics, Vol 250. New York: Springer, 2009

[6] Huang Q. Boundedness of iterated spherical average on modulation spaces. Nonlinear Analysis 2020; 199: 111968

[7] MiaoC. Harmonic Analysis and Its Applications on Partial Differential Equations. Beijing: Science Press, 1999 (in Chinese)

[8] Pearson K. The problem of random walk. Nature 1905; 72: 342

[9] Stein E M. Maximal functions: spherical means. Proc Nat Acad Sci USA 1976; 73(7): 2174–2175

[10] SteinE M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton N J: Princeton Univ Press, 1993

[11] TriebelH. Theory of Function Spaces, Monographs in Mathematcis, Vol 78. Basel: Birkhäuser, 1983

[12] WangBHuoZHaoCGuoZ. Harmonic Analysis Method for Nonlinear Evolution Equations I. Hackensack N J: World Scientific, 2011

RIGHTS & PERMISSIONS

Higher Education Press 2023