Journal of Inequalities and Applications 
Volume 2 (1998), Issue 4, Pages 325-356
doi:10.1155/S1025583498000216

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

R. C. Brown

 Department of Mathematics, University of Alabama, Tuscaloosa 35487-0350, AL, USA
Received 19 September 1997; Revised 25 November 1997

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 n. 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.