Riesz means on locally symmetric spaces

TL;DR

This paper proves almost everywhere convergence of Riesz means for functions in L^p spaces on certain rank one locally symmetric spaces, extending harmonic analysis techniques to these geometric settings.

Contribution

It establishes convergence criteria for Riesz means on rank one locally symmetric spaces, a novel extension of classical harmonic analysis results.

Findings

Riesz means converge almost everywhere for functions in L^p with specified conditions.

Convergence holds for Riesz order z with real part exceeding a critical threshold.

Results apply to a class of n-dimensional rank one locally symmetric spaces.

Abstract

We prove that for a certain class of $n$ dimensional rank one locally symmetric spaces, if $f \in L^p$, $1\leq p \leq 2$, then the Riesz means of order $z$ of $f$ converge to $f$ almost everywhere, for $\operatorname{Re}z> (n-1)(1/p-1/2).$