Lp bounds for Stein's spherical maximal operators

Abstract

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 $\mathrm{Re}\alpha \ge \max \{1/p-(n-1)/2,-(n-1)/p\}.$ When $n=2$, we prove that $\Vert {\mathfrak{M}}^{\alpha }f{\Vert }_{L^p({\mathbb{R}}^2)}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^2)}$ if $\mathrm{Re}\alpha >\max \{1/p-1/2,-1/p\}$, and hence the range of $\alpha$ is sharp in the sense the estimate fails for $\mathrm{Re}\alpha <\max \{1/p-1/2,-1/p\}.$

14 pages