We establish sharp inequalities of Trudinger-type
(with non-standard Young functions) on general domains Ω
with respect to measures ν, μ. Our results are new even when
is a Euclidean ball, and ν, μ are defined in terms of powers of distance to the boundary. In the Euclidean case, sharpness results are proved for the Young functions involved and the class of domains considered. New characterizations of QHBC domains are given
