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

TL;DR

This paper establishes precise embedding inequalities for reduced Sobolev spaces associated with Dirichlet problems with $L^1$ data, identifying optimal target spaces and their limiting behavior in dimension 2.

Contribution

It provides sharp embedding inequalities and determines the optimal target spaces for reduced Sobolev spaces related to $L^1$ data Dirichlet problems, including limiting cases in dimension 2.

Findings

Sharp embedding inequalities for reduced Sobolev spaces.

Identification of optimal target spaces for embeddings.

Limiting cases in dimension 2 related to Hansson-Brezis-Wainger spaces.

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 $W_0^{k,n/k}$.