Brennan conjecture for composition operators on Sobolev spaces

יום ב', 21/10/2013 - 14:00
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.