Lp bounds for Stein's spherical maximal operators

Let ${\mathfrak{M}}^{\alpha }$ be the spherical maximal operators of complex order $\alpha$ on ${\mathbb{R}}^n$. In this article we show that when $n\ge 2$, suppose \begin{eqnarray*} |{\frak M}^α f |{L^p({\mathbb R^n})} \leq C|f |{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $\alpha$ and $p\ge 2$, then we must have $\mathrm{R}\mathrm{e}\alpha \ge \max \{1\mathrm{/}p-(n-1)\mathrm{/}2,-(n-1)\mathrm{/}p\}\mathrm{.}$ When $n=2$, we prove that $\mathrm{\parallel }{\mathfrak{M}}^{\alpha }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^2)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^2)}$ if $\mathrm{R}\mathrm{e}\alpha >\max \{1\mathrm{/}p-1\mathrm{/}2,-1\mathrm{/}p\}$, and hence the range of $\alpha$ is sharp in the sense the estimate fails for $\mathrm{R}\mathrm{e}\alpha <\max \{1\mathrm{/}p-1\mathrm{/}2,-1\mathrm{/}p\}\mathrm{.}$