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

Abstract

The conformal mapping f (z) = (z+1) 2 from D onto the standard cardioid has a homeomorphic extension of finite distortion to entire R 2 . We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from D onto cardioid-type domains.

Singularities In L p-quasidisks

“…In this case the roles of p and q are interchanged. The reader interested in learning more about the conformal case f : D ≺ β onto −→ D is refer to [22]. Our proof of Theorem 1.11 is self-contained.…”

Sharp Pointwise Contraction of Mappings with Integrable Distortion Which Are Quasiconformal in a Disk

“…Note that the assumption on the domain Ω is quite general: It covers all cusp like domains, studied recently, for example, in [3,7,11], and allows for spiraling and deformation of the weak star domain. Furthermore, we note that rather surprisingly the right hand side of (3) is quite close, for any fixed p ≥ 1, to the classical quasiconformal bound by Morrey instead of the pointwise bound [8] for planar mappings with p-integrable distortion.…”

Singularities in L^p-quasidisks

“…In this case the roles of p and q are interchanged. The reader interested in learning more about the conformal case f : [22].…”