Density of smooth functions in Musielak–Orlicz–Sobolev spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$

We consider here Musielak–Orlicz–Sobolev (MOS) spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$, where Ω$\Omega$ is an open subset of Rd$\mathbb {R}^d$, k∈N,$k\in \mathbb {N,}$ and Φ$\Phi$ is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions CC∞(Ω)$C_C^\infty (\Omega)$ in Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$. One section is devoted to compare the various conditions on Φ$\Phi$ appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on Φ$\Phi$ we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions. One of the reasons is that in the process of proving density theorems, we do not use the fact that the Hardy–Littlewood maximal operator on Musielak–Orlicz space LΦ(Ω)$L^\Phi (\Omega)$ is bounded, a standard tool employed in density results for different types of Sobolev spaces. We show in particular that under some regularity assumptions on Φ$\Phi$, (A1) and Δ2$\Delta _2$ that are not sufficient for the maximal operator to be bounded, the space of CC∞(Rd)$C_C^\infty (\mathbb {R}^d)$ is dense in Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$. In the case of variable exponent Sobolev space Wk,p(·)(Rd)$W^{k,p(\cdot)}(\mathbb {R}^d)$, we obtain the similar density result under the assumption that Φ(x,t)=tp(x)$\Phi (x,t) = t^{p(x)}$, p(x)≥1$p(x)\ge 1$, t≥0$t\ge 0$, x∈Rd$x\in \mathbb {R}^d$, satisfies the log‐Hölder condition and the exponent function p$p$ is essentially bounded.