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.

Boundedness on inhomogeneous Lipschitz spaces of fractional integrals singular integrals and hypersingular integrals associated to non-doubling measures

“…The second application has appeared in [5] and it is an example for Theorem 3. In this paper the authors need to study the boundedness properties of the Restricted Beurling Transform, B Ω f = B(f χ Ω ), on Lip ε (Ω) where Ω is a bounded domain in R n with boundary of class C 1+ε , 0 < ε < 1.…”

A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

“…There is a lemma, which plays a crucial role in the studies of the smooth convolution Calderón–Zygmund operators with an even kernel. Lemma (Extra cancellation property, see [, Lemma 3] or ) Let B be an arbitrary ball in . Then .…”

“…The spaces are known to be invariant under certain smooth convolution Calderón–Zygmund operators (see, Peetre , Janson ). In the setting of function spaces defined on domains , the following result of Mateu, Orobitg and Verdera [, Main Lemma] is crucial. Theorem Let D be a bounded domain with ‐smooth boundary, .…”

Convergence of a numerical solver for an ℝ-linear Beltrami equation

“…It is reasonable to conjecture that then the solution u is also C δ -continuous of the same exponent δ as ν, α, β. See [11] for a result in this direction for a C-linear case.…”