Universal conformal weights on Sobolev spaces

TL;DR

This paper introduces universal conformal weights derived from the Jacobian of conformal maps to study Sobolev space embeddings into weighted Lebesgue spaces, establishing compactness results and applications to elliptic boundary value problems.

Contribution

It defines conformal weights based on the Jacobian of conformal maps and proves embedding and compactness results for Sobolev spaces into weighted Lebesgue spaces, extending to Brennan's conjecture.

Findings

Embeddings depend only on the conformal structure of the domain.

Proved compactness of Sobolev embeddings into weighted Lebesgue spaces.

Extended results to broader Sobolev spaces using Brennan's conjecture.

Abstract

The Riemann Mapping Theorem states existence of a conformal homeomorphism $\varphi$ of a simply connected plane domain $\Omega\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of the paper we study embeddings of Sobolev spaces $\overset{\circ}{W_{p}^{1}}(\Omega)$ into weighted Lebesgue spaces $L_{q}(\Omega,h)$ with an {}"universal" weight that is Jacobian of $\varphi$ i.e. $h(z):=J(z,\varphi)=| \varphi'(z)|^2$. Weighted Lebesgue spaces with such weights depend only on a conformal structure of $\Omega$. By this reason we call the weights $h(z)$ conformal weights. In the second part of the paper we prove compactness of embeddings of Sobolev spaces $\overset{\circ}{W_{2}^{1}}(\Omega)$ into $L_{q}(\Omega,h)$ for any $1\leq q<\infty$. With the help of Brennan's conjecture we extend these results to Sobolev spaces $\overset{\circ}{W_{p}^{1}}(\Omega)$. In this case $q$ is not arbitrary and depends on $p$ and the summability exponent for Brennan's conjecture. Applications to elliptic boundary value problems are demonstrated in the last part of the paper.