  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*}
