A density property for fractional weighted Sobolev spaces

TL;DR

This paper proves that functions in fractional weighted Sobolev spaces can be approximated by smooth, compactly supported functions despite the challenges posed by non-translation-invariant weights.

Contribution

It establishes a density property for fractional weighted Sobolev spaces, addressing the nonlocal and non-translation-invariant weight challenges.

Findings

Any function in the space can be approximated by smooth compactly supported functions.

The density property holds despite non-translation-invariant weights.

The result advances understanding of fractional weighted Sobolev spaces.

Abstract

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.