Optimal limiting embeddings for Δ-reduced Sobolev spaces in L1

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson–Brezis–Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W0k,n/k.