$L^p\to L^q$ 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 $\hat{I}\pm$ on ${\mathbb{R}}^n$: $${\mathfrak{M}}^{\hat{I}}{\pm }_{[1,2]}f(x)=\underset{t\in [1,2]}{\sup }|\frac{1}{\hat{I}“(\hat{I}\pm )}{\int }_{|y|\le 1}{\left(1-|y{|}^2\right)}^{\hat{I}\pm -1}f(x-ty)dy|.$$ We show that when $n\ge 2$, suppose $\Vert {\mathfrak{M}}^{\hat{I}}{\pm }_{[1,2]}f{\Vert }_{L^q({\mathbb{R}}^n)}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$ holds for some $\hat{I}\pm \in \mathbb{C}$, $p,q\ge 1$, then we must have that $q\ge p$ and $$\mathrm{Re}\hat{I}\pm \ge \ddot{I}{ƒ}_n(p,q):=\max \left\{\frac{1}{p}-\frac{n}{q},\frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak{M}}^{\hat{I}}{\pm }_{[1,2]}$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ provided that $q\ge p$ and $\mathrm{Re}\hat{I}\pm >\ddot{I}{ƒ}_2(p,q)$ for $n=2$; and $\mathrm{Re}\hat{I}\pm >\max \left\{\ddot{I}{ƒ}_n(p,q),1/(2p)-(n-2)/(2q)-(n-1)/4\right\}$ for $n>2$. The range of $\hat{I}\pm ,p$ and $q$ is almost optimal in the case either $n=2$, or $\hat{I}\pm =0$, or $(p,q)$ lies in some regions for $n>2$.

14 pages, 1 figures