An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

Given an arbitrary planar $\mathrm{\infty }$-harmonic function $u$, for each $\alpha >0$ we establish a quantitative local $W^{1,2}$-estimate of $\mathrm{\mid }Du{\mathrm{\mid }}^{\alpha }$, which is sharp as $\alpha \to 0$. 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 $\mathrm{\infty }$-harmonic functions in whole plane.