Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings

Abstract

In this paper we discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. A particular example are those of the form f ( z ) χ Ω ( z ) , where Ω is a bounded domain with boundary of class C 1 + ε and f a function in Lip ( ε , Ω ) satisfying ‖ f ‖ ∞ < 1 . An important point is that there is no restriction whatsoever on the Lip ( ε , Ω ) norm of f besides the requirement on Beltrami coefficients that the supremum norm be less than 1. The crucial fact in the proof is the extra cancellation enjoyed by even homogeneous Calderón–Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far reaching consequences. An application to a Lipschitz regularity result for solutions of second order elliptic equations in divergence form in the plane is presented.