A remark on convergence almost-everywhere of eigenfunction expansions of elliptic operators

TL;DR

This paper introduces a simple method to improve convergence theorems for eigenfunction expansions of elliptic operators, including new results on spherical Fourier series convergence for smooth functions.

Contribution

It proposes a novel, straightforward approach to estimate the maximal operator in L1, enhancing existing convergence theorems for eigenfunction expansions of elliptic operators.

Findings

Improved convergence theorems for eigenfunction expansions

New results on spherical partial sums of Fourier series

Enhanced understanding of convergence almost-everywhere

Abstract

In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.