Lp bounds for Stein's spherical maximal operators

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