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

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 $f\in L^1(G)$.