$L^p$ maximal bound and Sobolev regularity of two-parameter averages over tori

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to {\mathcal{A}}_t^{s}f$ over the two-parameter family of tori ${\mathbb{T}}_t^s:=\{((t+s\cos \theta )\cos \phi ,(t+s\cos \theta )\sin \phi ,s\sin \theta ):\theta ,\phi \in [0,2\pi )\}$ 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 $p>2$. We also obtain $L^p$–$L^q$ estimates for the local maximal operator on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operators $f\to {\mathcal{A}}_t^{s}f$ and $f\to {\mathcal{A}}_t^{c_{0}t}f$. For the purpose, we make use of Bourgain–Demeter’s decoupling inequality for the cone and Guth–Wang–Zhang’s local smoothing estimates for the $2$ dimensional wave operator.