$L^p\to L^q$ estimates for Stein's spherical maximal operators

Abstract: In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{\alpha -1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $|{\mathfrak M}^{\alpha}_{[1,2]} f |{L^q({\mathbb R^n})} \leq C|f |{L^p({\mathbb R^n})}$ holds for some $\alpha\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,\alpha\geq \sigma_n(p,q):=\max\left{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right}.$$ Conversely, we show that ${\mathfrak M}^\alpha_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,\alpha>\sigma_2(p,q)$ for $n=2$; and ${\rm Re}\,\alpha>\max\left{\sigma_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right}$ for $n>2$. The range of $\alpha,p$ and $q$ is almost optimal in the case either $n=2$, or $\alpha=0$, or $(p,q)$ lies in some regions for $n>2$.