Some embeddings of weighted sobolev spaces on finite measure and quasibounded domains

• Main Author: Brown RC
• Format: Article
• Language: English
• Published: SpringerOpen 1998-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Weighted Sobolev spaces; Continuous and compact embeddings; Regularity conditions; Minkowski dimension; Weighted Poincaré inequalities
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2/685839

<p/> <p>We show that several of the classical Sobolev embedding theorems extend in the case of weighted Sobolev spaces to a class of quasibounded domains which properly include all bounded or finite measure domains when the weights have an arbitrarily weak singularity or degeneracy at the boundary. Sharper results are also shown to hold when the domain satisfies an integrability condition which is equivalent to the Minkowski dimension of the boundary being less than <inline-formula><graphic file="1029-242X-1998-685839-i1.gif"/></inline-formula>. We apply these results to derive a class of weighted Poincar&#233; inequalities which are similar to those recently discovered by Edmunds and Hurri. We also point out a formal analogy between one of our results and an interpolation theorem of Cwikel.</p>