Moser-Trudinger trace inequalities

Sharp constants in exponential inequalities involving a general class of measures in domains Ω⊂Rn are exhibited in the limiting case of the Sobolev embedding theorem. A comprehensive approach is presented yielding, as special instances, trace inequalities on ∂Ω, on smooth submanifolds of Ω of arbitrary dimension, and also on fractal subsets of Ω, and recovering, in particular, the classical Moser-Trudinger inequality.