$L^p$ → $L^q$ Estimates for Stein’s Spherical Maximal Operators (Analysis in Theory and Applications, 2026)
In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on $\mathbb{R}^n$:$\mathfrak{M}_{[1,2]}^{α} f(x) = \sup\limits_{t \in [1,2]} \left| \frac{1}{Γ(α)} \int_{|y| ≤ 1} \left( 1 - |y|^2 \right)^{α - 1} f(x - ty) dy \right|.$We show that when $n ≥ 2$, suppose $\|\mathfrak{M}_{[1,2]}^{α} f\|_{L^q(\mathbb{R}^n)} ≤ C \|f\|_{L^p(\mathbb{R}^n)}$ holds for some $α \in \mathbb{C}$, $p, q ≥ 1$, then we must have that $q ≥ p$ and$$\operatorname{Re} α ≥ σ_n(p, q) := \max \left\{ \frac{1}{p} - \frac{n}{q},\; \frac{n+1}{2p} - \frac{n-1}{2} \left( \frac{1}{q} + 1 \right),\; \frac{n}{p} - n + 1 \right\}.$$Conversely, we show that $\mathfrak{M}_{[1,2]}^{α}$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ provided that $q ≥ p$ and $\operatorname{Re} α > σ_2(p, q)$ for $n = 2$; and$\operatorname{Re} α > \max \left\{ σ_n(p, q),\; 1/(2p) - (n-2)/(2q) - (n-1)/4 \right\}$for $n > 2$. The range of $α$, $p$ and $q$ is almost optimal in the case when either $n = 2$, or $α = 0$, or $(p, q)$ lies in certain regions for $n > 2$.
Keywords: spherical maximal operators; $L^p$-improving estimates; wave equation; local smoothing estimates