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

TL;DR

This paper investigates $L^p$ to $L^q$ bounds for a complex order modification of Stein's spherical maximal operator, establishing necessary and sufficient conditions on parameters for boundedness in various dimensions.

Contribution

It provides new bounds and conditions for the boundedness of the modified spherical maximal operator, extending previous results to complex orders and different dimensions.

Findings

Necessary conditions: $q \\geq p$ and ${\rm Re}\alpha \geq \sigma_n(p,q)$.

Sufficient conditions: boundedness when ${\rm Re}\alpha$ exceeds certain thresholds depending on $n$, $p$, and $q$.

Range of parameters is nearly optimal in key cases.

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 $n>2$.