Near Optimal $L^p\rightarrow L^q$ Estimates for Euclidean Averages Over Prototypical Hypersurfaces in $\mathbb{R}^3$

TL;DR

This paper determines the exact conditions on (p,q) for which local averages over certain hypersurfaces in three-dimensional space are bounded, using geometric methods applicable to a broad class of polynomial surfaces.

Contribution

It establishes the precise (p,q) range for restricted weak type estimates of Euclidean averages over polynomial hypersurfaces in R^3, employing non-oscillatory geometric techniques.

Findings

Identified the exact (p,q) range for boundedness of averages.

Developed geometric methods applicable to polynomial hypersurfaces.

Connected results to the broader class of real-analytic surfaces.

Abstract

We find the precise range of $(p,q)$ for which local averages along graphs of a class of two-variable polynomials in $\mathbb{R}^3$ are of restricted weak type $(p,q)$, given the hypersurfaces have Euclidean surface measure. We derive these results using non-oscillatory, geometric methods, for a model class of polynomials bearing a strong connection to the general real-analytic case.