On density of compactly supported smooth functions in fractional Sobolev spaces

Abstract We describe some sufficient conditions, under which smooth and compactly supported functions are or not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ ⊂ R d . The density property is closely related to lower upper Assouad codimension of boundary $$\Omega$$ also explicitly closure $$C_{c}^{\infty }(\Omega C c ∞ mild assumptions about geometry Finally, we prove a variant order Hardy inequality.