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

TL;DR

This paper establishes $L^p$ boundedness and Sobolev regularity results for two-parameter averaging operators over tori, using advanced decoupling and smoothing techniques, with sharp bounds and estimates.

Contribution

It proves the $L^p$ boundedness criterion for the maximal function over tori and derives sharp smoothing and local estimates using modern harmonic analysis tools.

Findings

Maximal function is bounded on $L^p$ if and only if $p>2$.

Established sharp $L^p$--$L^q$ estimates for local maximal operators.

Proved sharp local smoothing estimates for the averaging operators.

Abstract

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 $c_0t>s>0$ 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 $f\to \mathcal A_t^{c_0t} f$. 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.