  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.
