$L^p\to L^q$ estimates for Stein's spherical maximal operators
Naijia Liu, Minxing Shen, Liang Song, Lixin Yan
TL;DR
This work analyzes $L^p\to L^q$ bounds for a modified Stein spherical maximal operator of complex order ${\alpha}$, where the supremum is taken over the annulus $t\in[1,2]$. The authors derive necessary conditions $q\ge p$ and ${\rm Re}\,\alpha\ge {\sigma}_n(p,q)$, with ${\sigma}_n(p,q)$ defined via a max of three expressions, and establish sufficiency in several regimes: for $n=2$ when ${\rm Re}\,\alpha> {\sigma}_2(p,q)$, and for $n>2$ when ${\rm Re}\,\alpha> \max\{ {\sigma}_n(p,q), 1/(2p)-(n-2)/(2q)-(n-1)/4\}$. The approach decomposes the Fourier multiplier into a main oscillatory part related to half-wave propagators and an error term; necessity is demonstrated via three extremal test inputs, while sufficiency leverages local smoothing estimates for the wave equation, together with decoupling and interpolation. The results give near-optimal ranges in key cases and connect spherical maximal operators to local smoothing and wave propagation phenomena.
Abstract
In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on ${\mathbb R^n}$: $$ {\mathfrak M}^α_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over Γ(α) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{α-1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^α_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $α\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,α\geq σ_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak M}^α_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,α>σ_2(p,q)$ for $n=2$; and ${\rm Re}\,α>\max\left\{σ_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $α,p$ and $q$ is almost optimal in the case either $n=2$, or $α=0$, or $(p,q)$ lies in some regions for $n>2$.