Best constants in a borderline case of second order Moser type inequalities

We study optimal embeddings for the space of functions whose Laplacian \Delta u belongs to L^1(\Omega), where \Omega\subset\R^N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W^{2,1}(\Omega) in which the whole set of second order derivatives is considered. In particular, in the limiting Sobolev case, when N=2, we establish a sharp embedding inequality into the Zygmund space L_{exp}(\Omega). On one hand, this result enables us to improve the Brezis--Merle \cite{BM} regularity estimate for the Dirichlet problem \Delta u=f(x)\in L^1(\Omega), u=0 on \partial\Omega; on the other hand, it represents a borderline case of D.R. Adams'' \cite{DRA} generalization of Trudinger-Moser type inequalities to the case of higher order derivatives. Extensions to dimension N\geq are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions