Some problems of summability of spectral expansions connected with Laplace operator on sphere

TL;DR

This paper investigates the summability and convergence of Fourier-Laplace series on spheres, focusing on localization problems and employing Cesàro means and maximal operator estimates to address these issues.

Contribution

It introduces new estimations for maximal operators related to Fourier-Laplace series on spheres, advancing understanding of their summability properties.

Findings

Cesàro means improve summability of Fourier-Laplace series

Maximal operator estimates are linked to Hardy-Littlewood's maximal function

Results apply to localization problems in spherical harmonic analysis

Abstract

Solution of some boundary value problems and initial problems in unique ball leads to the convergence and sumability problems of Fourier series of given function by eigenfunctions of Laplace operator on a sphere - spherical harmonics. Such a series are called as Fourier-Laplace series on sphere. There are a number of works devoted investigation of these expansions in different topologies and for the functions from the various functional spaces. In the present work we consider only localization problems in both usual and generalized (almost everywhere localization) senses in the classes of summable functions. We use Chezaro means of the partial sums for study of summability problems. In order to prove the main theorems we obtaine estimations for so called maximal operator estimating it by Hardy-Littlwood's maximal function. Significance of this function is that it majors of many important operators of mathematical physics. For instance, Hardy-Littlewood's maximal function majors Poisson's integral in the space.