We study optimal embeddings for the space of functions whose Laplacian belongs to L1(Ω), where Ω⊂RN is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω) 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 Lexp(Ω). This result enables us to improve the Brezis-Merle regularity estimate for the Dirichlet problem Δu=f(x)∈L1(Ω), u=0 on ∂Ω. We then study the operator associated to this problem, the 1-biharmonic operator.