Extra cancellation of even Calderón-Zygmund operators and quasiconformal mappings

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) χΩ (z), where Ω is a bounded domain with boundary of class C1 + ε and f a function in Lip (ε, Ω) satisfying {norm of matrix} f {norm of matrix}∞ < 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. © 2009 Elsevier Masson SAS. All rights reserved.