Sparse domination and $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with curvature

TL;DR

This paper develops a framework for sparse domination of maximal functions along finite type curves and hypersurfaces, establishing new $L^{p} ightarrow L^{q}$ bounds considering non-isotropic dilations and curvature vanishing.

Contribution

It introduces a generic scheme for sparse domination bounds and derives $L^{p} ightarrow L^{q}$ estimates for localized maximal functions with non-isotropic dilations.

Findings

Sparse domination bounds for global maximal functions.

$L^{p} ightarrow L^{q}$ bounds for localized maximal functions with finite type curvature.

Weighted inequalities for global maximal functions.

Abstract

In this paper, we study maximal functions along some finite type curves and hypersurfaces. In particular, various impacts of non-isotropic dilations are considered. Firstly, we provide a generic scheme that allows us to deduce the sparse domination bounds for global maximal functions under the assumption that the corresponding localized maximal functions satisfy the $L^{p}$ improving properties. Secondly, for the localized maximal functions with non-isotropic dilations of curves and hypersurfaces whose curvatures vanish to finite order at some points, we establish the $L^{p}\rightarrow L^{q}$ bounds $(q >p)$. As a corollary, we obtain the weighted inequalities for the corresponding global maximal functions, which generalize the known unweighted estimates.