Fractional Sobolev spaces with power weights

Abstract

We investigate the form of the closure of the smooth, compactly supported functions C ∞ c (Ω) in the weighted fractional Sobolev space W s,p; w,v (Ω) for bounded Ω. We focus on the weights w, v being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of Ω.For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.Contents 1 p and the weighted fractional Sobolev space as W s,p; w,v (Ω) = f ∈ L p (Ω) : [f ] W s,p; w,v (Ω) < ∞ .