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 (Ann Inst Fourier Grenoble 24(1):149–172, 1974) 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

$$\begin{aligned} \limsup _{R\rightarrow \infty } \big |S_{R}^{(n-1)/2}(f)(x)\big |=\infty ,\quad \ a.e.\ x\in G. \end{aligned}$$

This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324–328, 1923) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140–170, 1961) 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 $f\in L^1(G)$.