$$L^p$$ 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

$$\begin{aligned} \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^n})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^n})} \end{aligned}$$

holds for some \(\alpha \) and \(p\ge 2\), then we must have that \(\textrm{Re}\,\alpha \ge \max \{1/p-(n-1)/2,\ -(n-1)/p \}.\) In particular, 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 \(\textrm{Re}\ \! \alpha >\max \{1/p-1/2,\ -1/p\}\), and consequently the range of \(\alpha \) is sharp in the sense that the estimate fails for \(\textrm{Re}\ \alpha <\max \{1/p-1/2, -1/ p\}.\)