On $L^p$-improving for averages associated to mixed homogeneous polynomial hypersurfaces in $\mathbb{R}^3$

TL;DR

This paper derives $L^p-L^q$ estimates for averaging operators linked to mixed homogeneous polynomial hypersurfaces in three-dimensional space, based on their geometric and algebraic properties.

Contribution

It provides new $L^p-L^q$ bounds for these operators, connecting geometric features like homogeneity and curvature to harmonic analysis estimates.

Findings

Established $L^p-L^q$ estimates for mixed homogeneous polynomial hypersurfaces.

Connected geometric properties to the boundedness of averaging operators.

Enhanced understanding of the role of curvature and homogeneity in harmonic analysis.

Abstract

We establish $L^p-L^q$ estimates for averaging operators associated to mixed homogeneous polynomial hypersurfaces in $\mathbb{R}^3$. These are described in terms of the mixed homogeneity and the order of vanishing of the polynomial hypersurface and its Gaussian curvature transversally to their zero sets.