About summability of Fourier-Laplace series

TL;DR

This paper investigates the conditions under which Fourier-Laplace series converge almost everywhere, establishing new summability criteria and maximal operator estimates for Riesz means beyond the critical index.

Contribution

It introduces new summability methods for Laplace series based on spectral properties of the self-adjoint Laplace-Beltrami operator, extending convergence results.

Findings

Established sufficient conditions for summability of Fourier-Laplace series.

Derived maximal operator estimates for Riesz means of order greater than the critical index.

Constructed new summability methods based on spectral expansion properties.

Abstract

In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we established the estimation for maximal operator of the Riesz means. Note that when order $\alpha$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series, which based on spectral expansions property of self-adjoint Laplace-Beltrami operator on the unit sphere