Lp maximal bounds and Sobolev regularity of two-parameter averages over tori

Abstract

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\mapsto A_t^{s}f$ over the two-parameter family of tori$T_t^s:=\left\{\right((t+s\cos \theta )\cos \varphi ,(t+s\cos \theta )\sin \varphi ,s\sin \theta ):\theta ,\varphi \in [0,2\pi \left)\right\}$ with $c_{0}t>s>0$ for some $c_0\in (0,1)$. We prove that the associated (two-parameter) maximal function is bounded on $L^p$ if and only if $p>2$. Also, we obtain $L^p$–$L^q$ bounds for the local maximal operator on a sharp range of $p,q$. Furthermore, sharp smoothing estimates are obtained including the local smoothing estimates for the operators $f\mapsto A_t^{s}f$ and $f\mapsto A_t^{c_{0}t}f$. For these purposes, we make use of Bourgain–Demeter's decoupling inequality and Guth–Wang–Zhang's local smoothing estimate for the 2-dimensional wave operator.