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 $n\ge 2$ and $M^{\hat{I}}\pm$ is bounded on $L^p({\mathbb{R}}^n)$ for some $\hat{I}\pm \in \mathbb{C}$ and $p\ge 2$, then we have \begin{align*} \operatorname{Re} α\geq \max\left\{\frac{1-n}{2}+\frac{1}{p},\frac{1-n}{p}\right\}. \end{align*}

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