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

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.