Hessian determinants and averaging operators over surfaces in ${\mathbb R}^3$

TL;DR

This paper establishes optimal Sobolev and Lebesgue space improvement results for local averaging operators over real analytic surfaces in three dimensions, utilizing advanced oscillatory integral and resolution of singularities techniques.

Contribution

The paper introduces a novel approach combining oscillatory integral methods with resolution of singularities to achieve maximal derivative decay in surface measure Fourier transforms.

Findings

Proves Sobolev improvement theorems for averaging operators in ${\mathbb R}^3$

Achieves optimal $(p,s)$ boundedness ranges up to endpoints

Develops a method to attain the maximum possible 1 derivative decay

Abstract

We prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ Sobolev improvement theorems for local averaging operators over real analytic surfaces in ${\mathbb R}^3$. For most such operators, in a sense made precise in the paper, the set of $(p,s)$ for which we prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ boundedness is optimal up to endpoints. Using an interpolation argument in conjunction with these $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ results we obtain an $L^p({\mathbb R}^3)$ to $L^q({\mathbb R}^3)$ improvement theorem, and the set of exponents $(p,q)$ obtained will also usually be optimal up to endpoints. The advantage the methods of this paper have over those of the author's earlier papers is that the oscillatory integral methods of the earlier papers, closely tied to the Van der Corput lemma, allow one to only prove 1/2 of a derivative of surface measure Fourier transform decay, while the methods of this paper, when combined with appropriate resolution of singularities methods, allow one to go up to the maximum possible 1 derivative. This allows us to prove the stronger sharp up to endpoints results.