On Beltrami equations and Hoelder regularity

Abstract

We estimate the Hoelder exponent $\alpha$ of solutions to the Beltrami equation $\dbar f=\mu\de f$, where the Beltrami coefficient satisfies $\|\mu\|_\infty<1$. Our estimate improves the classical estimate $\alpha\ge\|K_\mu\|^{-1}$, where $K_\mu=(1+|\mu|)/(1-|\mu|)$, and it is sharp, in the sense that it is actually attained in a class of mappings which generalize the radial stretchings. Some other properties of such mappings are also provided.