Brennan conjecture for composition operators on Sobolev spaces

We show that Brennan's conjecture about integrability of

derivatives of conformal homeomorphisms is equivalent to boundedness

of composition operators on homogeneous Sobolev spaces $L^{1,p}$.

This result is used for description of embedding operators of

homogeneous Sobolev spaces $L^{1,p}$ into weighted Lebesgue spaces

with so-called "conformal weights" induced by the conformal

homeomorphisms of simply connected plane domains to the unit disc.

Applications to elliptic boundary value problems will be discussed.