Fractional Sobolev spaces with power weights

TL;DR

This paper studies the structure of weighted fractional Sobolev spaces with power weights near domain boundaries, revealing how boundary geometry influences function space closures and seminorm equivalences.

Contribution

It characterizes the closure of smooth functions in weighted fractional Sobolev spaces with power weights based on boundary codimension and establishes seminorm comparability.

Findings

Closure of smooth functions depends on boundary codimension.

Weighted fractional Gagliardo seminorm is comparable to truncated seminorm.

Results apply to domains with boundary of specific Assouad codimension.

Abstract

We investigate the form of the closure of the smooth, compactly supported functions $C_{c}^{\infty}(\Omega)$ in the weighted fractional Sobolev space $W^{s,p;\,w,v}(\Omega)$ for bounded $\Omega$. 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 $\Omega$. For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.