On Beltrami equations and Hoelder regularity

Abstract

We estimate the Hoelder exponent $α$ of solutions to the Beltrami equation $\dbar f=μ\de f$, where the Beltrami coefficient satisfies $\|μ\|_\infty<1$. Our estimate improves the classical estimate $α\ge\|K_μ\|^{-1}$, where $K_μ=(1+|μ|)/(1-|μ|)$, 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.