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

Abstract

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 . We apply these results to derive a class of weighted Poincaré 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.

Keywords

• Weighted Sobolev spaces
• Continuous and compact embeddings
• Regularity conditions
• Minkowski dimension
• Weighted Poincaré inequalities