A note on generalized spherical maximal means

Abstract

The goal of this note is to provide an alternative proof of Theorem 1.1 (i) in [4], that is, if $n\geq 2$ and $M^{\alpha}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for some $\alpha\in \mathbb{C}$ and $p\geq 2$, then we have \begin{align*} \operatorname{Re} \alpha\geq \max\left\{\frac{1-n}{2}+\frac{1}{p},\frac{1-n}{p}\right\}. \end{align*}