We prove that a function belonging to a fractional Sobolev space $L^{\alpha ,p}\left(ℝⁿ\right)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,\lambda }\left(ℝⁿ\right)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.

We prove several results concerning density of $C_0^{\infty }$, behaviour at infinity and integral representations for elements of the space $L^{m,p}=⨍|{\nabla }^m⨍\in L^p$.