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 $$\underset{R\to \infty }{\mathrm{limsup}}|S_R^{(n-1)/2}(f)(x)|=\infty ,\text{a.e.}x\in G.$$ 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 ${\mathbb{T}}^n,n\ge 2$. We also study localization properties of the Bochner-Riesz mean $S_R^{(n-1)/2}(f)$ for $f\in L^1(G)$.

arXiv comment22 pages; remarks on localization added