Best constants for Moser type inequalities in Zygmund spaces

We first survey some recent results on optimal embeddings for the space of functions whose \Delta u\in L^1(\Omega), where \Omega\subsetR^2 is a bounded domain. The target space in the embeddings turns out to be a Zygmund space and the best constants are explicitly known. Remarkably, the best constant in the case of zero boundary data is twice the best constant in the case of compactly supported functions. Then, following the same strategy, we establish a new version of the celebrated Trudinger--Moser inequality, in the Zygmund space Z_0^{1/2}(\Omega), and we prove that, in contrast to the Moser case, here the best embedding constant is not attained.