Universal conformal weights on Sobolev spaces

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.