A Moser type inequality in Zygmund spaces without boundary conditions

We survey old and new results about the so-called limiting Sobolev case for the embedding of the space W^{1,n}_0(Ω) into suitable spaces of functions having exponential summability. In particular, we discuss a new notion of criticality with respect to attainability of the best constant in the related embedding inequalities and the connection with existence and nonexistence of solutions to boundary value problems, in which Moser’s functions are cast in a new framework. Then, we prove a new version of Moser’s inequality in Zygmund spaces with respect to the full Sobolev norm and without boundary conditions.