Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings.pdfVIP

a r X i v : 0 8 0 2 .1 1 8 5 v 1 [ m a t h .C A ] 8 F e b 2 0 0 8 Extra cancellation of even Caldero?n-Zygmund operators and Quasiconformal mappings Joan Mateu, Joan Orobitg and Joan Verdera Abstract In this paper we discuss a special class of Beltrami coefficients whose asso- ciated quasiconformal mapping is bilipschitz. These are of the form f(z)χ?(z), where ? is a bounded domain with boundary of class C1+ε and f a function in Lip(ε,?) satisfying ‖f‖∞ 1. An important point is that there is no restriction whatsoever on the Lip(ε,?) 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 Caldero?n- 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. 1 Introduction Consider the Beltrami equation ?Φ ?z (z) = μ(z) ?Φ ?z (z) , z ∈ C , (1) where μ is a Lebesgue measurable function on the complex plane C satisfying ‖μ‖∞ 1 . According to a remarkable old theorem of Morrey [M] there exists an essentially unique function Φ in the Sobolev space W 1,2loc (C) (functions with first order derivatives locally in L2) which satisfies (1) almost everywhere and is a homeo- morphism of the plane. These functions are called quasiconformal. It turns out that Φ may change drastically the Hausdorff dimension of sets. Indeed, sets of arbitrarily small positive Hausdorff dimension may be mapped into sets of Hausdorff dimension as close to 2 as desired (and the other way around by the inverse mapping). There has been much hard and brilliant work in understanding how Φ distorts sets (see, for instance [As], and the references given there). In [R] one gives geometric con- ditions which are necessary and sufficient for Φ being bilipschitz, but which do not involve the Beltrami coefficient μ. In fact, it is widely accepted that the p