Optimal limiting embeddings for Δ-reduced Sobolev spaces in $ L^{1} $

Luigi Fontana; Carlo Morpurgo

doi:10.1016/j.anihpc.2013.02.007

JournalsaihpcVol. 31, No. 2pp. 217–230

Optimal limiting embeddings for Δ-reduced Sobolev spaces in $L^{1}$

Luigi Fontana
Dipartimento di Matematica e Applicazioni, Universitá di Milano-Bicocca, Via Cozzi, 53, 20125 Milano, Italy
Carlo Morpurgo
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

$Optimal limiting embeddings for Δ-reduced Sobolev spaces in $ L^{1} $ cover$

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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 $W_{0}^{k, n / k}$ .

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Luigi Fontana, Carlo Morpurgo, Optimal limiting embeddings for Δ-reduced Sobolev spaces in $L^{1}$ . Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, pp. 217–230

DOI 10.1016/J.ANIHPC.2013.02.007