On almost everywhere divergence of Bochner-Riesz means on compact Lie groups

TL;DR

This paper demonstrates that at the critical index, the Bochner-Riesz means on compact Lie groups can diverge almost everywhere, extending classical divergence results to a broader geometric setting.

Contribution

It establishes almost everywhere divergence of Bochner-Riesz means at the critical index on compact Lie groups, generalizing known results from Fourier analysis on simpler domains.

Findings

Existence of functions with divergence at the critical index

Extension of divergence results to compact Lie groups

Analysis of localization properties of Bochner-Riesz means

Abstract

Let $G$ be a connected, simply connected, compact semisimple Lie group of dimension $n$. It has been shown by Clerc \cite{Clerc1974} that, for any $f\in L^1(G)$, the Bochner-Riesz mean $S_R^\delta(f)$ converges almost everywhere to $f$, provided $\delta>(n-1)/2$. In this paper, we show that, at the critical index $\delta=(n-1)/2$, there exists an $f\in L^1(G)$ such that $$\limsup_{R\rightarrow\infty} \big|S_{R}^{(n-1)/2}(f)(x)\big|=\infty, \ \text{a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov \cite{Kolmogoroff1923} for Fourier series on the circle, and a result of Stein \cite{Stein1961} for Bochner-Riesz means on the tori $\mathbb T^{n}, n\geq 2$. We also study localization properties of the Bochner-Riesz mean $S_{R}^{(n-1)/2}(f)$ for $f\in L^1(G)$.