Optimal limiting embeddings for $\Delta$-reduced Sobolev spaces in $L^1$

Abstract

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with $L^1$ 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 $W_0^{k,n/k}$.

In v3 the proof of the sharpness of (25) and (29) was corrected, and minor other changes and corrections were added. To appear in Ann. Inst. H. Poincaré Anal. Non Linéaire