$L^p$ → $L^q$ Estimates for Stein’s Spherical Maximal Operators

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on $\mathbb{R}^n$: $\mathfrak{M}_{[1,2]}^{\alpha} f(x) = \sup\limits_{t \in [1,2]} \left| \frac{1}{\Gamma(\alpha)} \int_{|y| \leq 1} \left( 1 - |y|^2 \right)^{\alpha - 1} f(x - ty) dy \right|.$ We show that when $n \geq 2$, suppose $\|\mathfrak{M}_{[1,2]}^{\alpha} f\|_{L^q(\mathbb{R}^n)} \leq C \|f\|_{L^p(\mathbb{R}^n)}$ holds for some $\alpha \in \mathbb{C}$, $p, q \geq 1$, then we must have that $q \geq p$ and $$\operatorname{Re} \alpha \geq \sigma_n(p, q) := \max \left\{ \frac{1}{p} - \frac{n}{q},\; \frac{n+1}{2p} - \frac{n-1}{2} \left( \frac{1}{q} + 1 \right),\; \frac{n}{p} - n + 1 \right\}.$$ Conversely, we show that $\mathfrak{M}_{[1,2]}^{\alpha}$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ provided that $q \geq p$ and $\operatorname{Re} \alpha > \sigma_2(p, q)$ for $n = 2$; and $\operatorname{Re} \alpha > \max \left\{ \sigma_n(p, q),\; 1/(2p) - (n-2)/(2q) - (n-1)/4 \right\}$ for $n > 2$. The range of $\alpha$, $p$ and $q$ is almost optimal in the case when either $n = 2$, or $\alpha = 0$, or $(p, q)$ lies in certain regions for $n > 2$.