On weighted Sobolev spaces

Abstract

We study density and extension problems for weighted Sobolev spaces on bounded (∈, δ) domains script D sign when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of script D sign and when it is in the Muckenhoupt Ap class locally in script D sign. Moreover, when the weights wi(x) are of the form dist(x. Mi)αi, αi, ∈ ℝ, Mi ⊂ script D sign that are doubling, we are able to obtain some extension theorems on (∈ ∞) domains.