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

Luigi Fontana; Carlo Morpurgo

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

Functional Analysis 2013-03-26 v3

Authors: Luigi Fontana , Carlo Morpurgo

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

Keywords

sobolev spaces metric embeddings sobolev inequalities

Cite

@article{arxiv.1202.1133,
  title  = {Optimal limiting embeddings for $\Delta$-reduced Sobolev spaces in $L^1$},
  author = {Luigi Fontana and Carlo Morpurgo},
  journal= {arXiv preprint arXiv:1202.1133},
  year   = {2013}
}

Comments

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. Poincar\'e Anal. Non Lin\'eaire

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