Maximal functions related to homogeneous hypersurfaces in $\mathbb{R}^3$

TL;DR

This paper investigates the boundedness of maximal functions related to homogeneous polynomial hypersurfaces in three-dimensional space, establishing optimal $L^p$ to $L^q$ bounds and weighted inequalities.

Contribution

It provides explicit $L^p$ bounds depending on hypersurface height and curve type, including global estimates without transversality assumptions.

Findings

Established optimal $L^p$ to $L^q$ boundedness regions.

Derived weighted norm inequalities for global maximal functions.

Obtained optimal $L^p$ estimates without transversality conditions.

Abstract

We study maximal functions related to homogeneous polynomial hypersurfaces in $\mathbb{R}^3$. In a sense made precise in this paper, the region of $(p,q)$ for which we obtain $L^p\rightarrow L^q$ boundedness is optimal up to the endpoints for the corresponding local maximal operators. The boundedness exponents depend explicitly on both the height of the hypersurface and the type of the curve determined by the level set. As a corollary, we obtain $L^p$-estimates and weighted norm inequalities for the associated global maximal functions. Moreover, we also obtain optimal $L^{p}$-estimates for the global maximal operators associated with homogeneous polynomial hypersurfaces without transversality condition in $\mathbb{R}^{3}$.