Riesz means on symmetric spaces

TL;DR

This paper proves almost everywhere convergence of Riesz means on non-compact symmetric spaces for functions in L^p, establishing conditions on the order of the means for convergence as the parameter grows.

Contribution

It establishes new convergence results for Riesz means on symmetric spaces, extending harmonic analysis techniques to non-compact settings.

Findings

Almost everywhere convergence for Riesz means when Re z exceeds a specific threshold.

Convergence holds for functions in L^p with 1 ≤ p ≤ 2.

Provides explicit conditions linking the order of Riesz means to the space dimension and p.

Abstract

Let $X$ be a non-compact symmetric space of dimension $n$. We prove that if $f\in L^{p}(X)$, $1\leq p\leq 2$, then the Riesz means $S_{R}^{z}\left( f\right)$ converge to $f$ almost everywhere as $R\rightarrow \infty $, whenever $\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) $.