# Optimal extensions of conformal mappings from the unit disk to cardioid-type domains

## TL;DR

This paper investigates the optimal regularity of finite distortion extensions of conformal mappings from the unit disk to cardioid-type domains, focusing on integrability properties of distortion and derivatives.

## Contribution

It generalizes existing results to a broader class of cardioid-type domains, analyzing the optimal regularity and integrability of these extensions and their inverses.

## Key findings

- Extensions have optimal regularity in terms of distortion integrability.
- Generalization from the standard cardioid to cardioid-type domains.
- Results characterize the regularity of both the mappings and their inverses.

## Abstract

The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ 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 $\mathbb{D}$ onto cardioid-type domains.