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

Abstract

We consider here Musielak–Orlicz–Sobolev (MOS) spaces $W^{k,\Phi }\left(\Omega \right)$, where $\Omega$ is an open subset of $R^d$, $k\in N,$ and $\Phi$ is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions $C_C^{\infty }\left(\Omega \right)$ in $W^{k,\Phi }\left(\Omega \right)$. One section is devoted to compare the various conditions on $\Phi$ appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on $\Phi$ 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 $L^{\Phi }\left(\Omega \right)$ 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 $\Phi$, (A1) and ${\Delta }_2$ that are not sufficient for the maximal operator to be bounded, the space of $C_C^{\infty }\left(R^d\right)$ is dense in $W^{k,\Phi }\left(\Omega \right)$. In the case of variable exponent Sobolev space $W^{k,p(·)}\left(R^d\right)$, we obtain the similar density result under the assumption that $\Phi (x,t)=t^{p\left(x\right)}$, $p\left(x\right)\ge 1$, $t\ge 0$, $x\in R^d$, satisfies the log-Hölder condition and the exponent function $p$ is essentially bounded.