On density of smooth functions in weighted fractional Sobolev spaces

TL;DR

This paper proves that smooth functions are dense in weighted fractional Sobolev spaces under mild conditions, extending classical approximation results to these nonlocal and weighted contexts.

Contribution

It establishes density of smooth functions in weighted fractional Sobolev spaces on arbitrary open sets, generalizing the Meyers--Serrin theorem to nonlocal and weighted settings.

Findings

Smooth functions are dense in weighted fractional Sobolev spaces.

Similar density results hold in non-weighted spaces with specific kernels.

Results extend classical approximation theorems to fractional and weighted contexts.

Abstract

We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel similar to $x\mapsto |x|^{-d-sp}$. One may consider the results to be a~version of the Meyers--Serrin theorem.