Compactly supported smooth functions are dense in Slobodeckij spaces

Truncation. Choose $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ with $0\le \eta \le 1$, equal to one on $B(0,1)$, and set ${\eta }_R(x)=\eta (x\mathrm{/}R)$ and $A_R=(1-{\eta }_R)g$. Dominated convergence gives $\mathrm{\parallel }{A}_R{\mathrm{\parallel }}_p\to 0$. Write $A_R(x)-A_R(y)=(1-{\eta }_R(x))(g(x)-g(y))+({\eta }_R(y)-{\eta }_R(x))g(y)$. The weighted integral of the first term's $p$-th power tends to zero by dominated convergence, dominated by the defining seminorm integrand of $g$. For the second, $\mathrm{\mid }{\eta }_R(x)-{\eta }_R(y)\mathrm{\mid }\le \min (2,\mathrm{\parallel }D\eta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}R)$; translating $h=x-y$ and scaling $h=Rz$ bounds its weighted integral by $C{R}^{-sp}\mathrm{\parallel }g{\mathrm{\parallel }}_p^p$, with $C<\mathrm{\infty }$ by [F9]. The inequality $\mathrm{\mid }a+b{\mathrm{\mid }}^p\le 2^{p-1}(\mathrm{\mid }a{\mathrm{\mid }}^p+\mathrm{\mid }b{\mathrm{\mid }}^p)$ now gives $[A_R{]}_{s,p}\to 0$, so ${\eta }_{R}g\to g$ in $W^{s,p}$.

Translation continuity. For $\phi \in W^{s,p}$ define $F(x,h)=(\phi (x+h)-\phi (x))\mathrm{\mid }h{\mathrm{\mid }}^{-(d+sp)\mathrm{/}p}$ off $h=0$, and zero there. Tonelli and translation invariance give $F\in L^p({\mathbb{R}}^{2d})$ with $\mathrm{\parallel }F{\mathrm{\parallel }}_p=[\phi {]}_{s,p}$. Translating only its $x$ coordinate gives $[\phi (\cdot -z)-\phi {]}_{s,p}=\mathrm{\parallel }F(\cdot -z,\cdot )-F{\mathrm{\parallel }}_{L^p({\mathbb{R}}^{2d})}\to 0$ by [F3] in dimension $2d$. Also this seminorm is at most $2[\phi {]}_{s,p}$ by [F2] and [F4].

Mollification of a compactly supported class. Let $\phi \in W^{s,p}$ be compactly supported and let ${\rho }_{\epsilon }$ be a standard nonnegative mollifier as in [F6]. Since ${\rho }_{\epsilon }$ has mass one, $({\rho }_{\epsilon }\ast \phi -\phi )(x)-({\rho }_{\epsilon }\ast \phi -\phi )(y)=\int {\rho }_{\epsilon }(z)[\phi (x-z)-\phi (x)-\phi (y-z)+\phi (y)]dz$, and Holder's inequality in the probability measure ${\rho }_{\epsilon }(z)dz$ gives $[{\rho }_{\epsilon }\ast \phi -\phi {]}_{s,p}^p\le \int {\rho }_{\epsilon }(z)[\phi (\cdot -z)-\phi {]}_{s,p}^{p}dz$ after integrating the pointwise $p$-th power estimate and exchanging the $z$- and $(x,y)$-integrals by Tonelli [F8]. By step 1.2 the integrand is bounded by $2^p[\phi {]}_{s,p}^p$ and tends to $0$ as $z\to 0$, while ${\int }_{\mathrm{\mid }z\mathrm{\mid }>\delta }{\rho }_{\epsilon }(z)dz\to 0$ for each $\delta >0$ by [F6]; hence $\int {\rho }_{\epsilon }(z)[\phi (\cdot -z)-\phi {]}_{s,p}^{p}dz\to 0$ as $\epsilon \downarrow 0$. Also $\mathrm{\parallel }{\rho }_{\epsilon }\ast \phi -\phi {\mathrm{\parallel }}_p\to 0$ by [F5]. Therefore ${\rho }_{\epsilon }\ast \phi \to \phi$ in $W^{s,p}({\mathbb{R}}^d)$, and each ${\rho }_{\epsilon }\ast \phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ by [F6].

Conclusion. Let $g\in W^{s,p}({\mathbb{R}}^d)$ and $\epsilon >0$. By step 1.1 choose $R$ with $\mathrm{\parallel }{\eta }_{R}g-g{\mathrm{\parallel }}_{W^{s,p}}<\epsilon \mathrm{/}2$. The class $\phi ={\eta }_{R}g$ is compactly supported, and step 2.1 supplies a mollifier scale $\delta >0$ with $\mathrm{\parallel }{\rho }_{\delta }\ast \phi -\phi {\mathrm{\parallel }}_{W^{s,p}}<\epsilon \mathrm{/}2$. Then ${\rho }_{\delta }\ast \phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ and $\mathrm{\parallel }g-{\rho }_{\delta }\ast \phi {\mathrm{\parallel }}_{W^{s,p}}<\epsilon$.