An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

TL;DR

This paper establishes sharp asymptotic Sobolev regularity results for planar infinity harmonic functions, providing new quantitative estimates and bounds that deepen understanding of their regularity and measure-theoretic properties.

Contribution

It introduces the first sharp local Sobolev estimates for |Du|^α as α approaches zero and analyzes the distributional determinant of infinity harmonic functions.

Findings

Quantitative local W^{1,2}-estimates for |Du|^α as α→0

Bounds on the distributional determinant as a Radon measure

W^{1,p}-estimates and L^p-Liouville property for infinity harmonic functions

Abstract

Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which is sharp as $\alpha\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\infty$-harmonic functions in whole plane.