Maximal averages associated to families of finite type surfaces

TL;DR

This paper establishes $L^p$ boundedness of maximal operators associated with finite type hypersurfaces in $ $, extending to variable coefficients and confirming conjectures relating Fourier decay and boundedness.

Contribution

It proves $L^p$ boundedness for maximal operators on finite type hypersurfaces, including variable coefficient cases, and verifies related conjectures about Fourier decay and boundedness.

Findings

Maximal operators are bounded on $L^p$ for $p>k$ for finite type hypersurfaces.

The same boundedness holds for variable coefficient maximal operators.

Confirmed conjectures linking Fourier decay rates to $L^p$ boundedness.

Abstract

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of finite type $k$ at $x_0 \in \mathbb{R}^n$, then the associated maximal operator is bounded on $L^p(\mathbb{R}^n)$ for $p>k.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the same $L^p-$ boundedness result for $p>k.$ We also discuss the consequence of this result. In particular, we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on $S$ and the $L^p-$ boundedness of the associated maximal operator $\mathcal{M}.$