Chord-arc curves and the Beurling transform

Abstract

We study the relation between the geometric properties of a quasicircle~$\Gamma$ and the complex dilatation~$\mu$ of a quasiconformal mapping that maps the real line onto~$\Gamma$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-\mu S)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves.Comment: 27 page

BMO Teichmüller spaces and their quotients with complex and metric structures

“…There is a long-standing open question about whether T c is connected or not. For a recent account to a related result, see Astala and González [3]. In this section, we prove a result concerning the distribution of T c in T b .…”

Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains

“…Let Ω ⊂ R 2 and Ω ′ ⊂ R 2 be domains. A homeomorphism f : Ω → Ω ′ is called K-quasiconformal if f ∈ W 1,2 loc (Ω, R 2 ) and if there is a constant K ≥ 1 such that |Df (z)| 2 ≤ KJ f (z) holds for L 2 -a.e. z ∈ Ω.…”

Causal sparse domination of Beurling maximal regularity operators

“…At the end of the present paper, we explain how our Theorem 5.1, in the presence of pointwise kernel bounds, sharpens the estimate of half of the maximal regularity operators in [3]. In the case of the Laplace equation in R 2 + , the maximal regularity operators S ± appearing are the two causal parts of the Beurling transform, in which case one of the key estimates from [3] independently have appeared in Astala-González [2]. We recall that the kernel k(x; y) = (x − y) −2 of this Beurling transform is symmetric and as a consequence its causal parts S ± are L 2 bounded.…”