Density of smooth functions in Musielak–Orlicz spaces

Abstract

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $C^\infty _c(\Omega )$ to be dense in Musielak–Orlicz spaces $L^\Phi (\Omega )$ where $\Omega$ is an open subset of ${\mathbb {R}}^d$. In particular, we prove that if $\Phi$ satisfies condition $\Delta _2$, the closure of $C^\infty _c(\Omega )\cap L^\Phi (\Omega )$ is equal to $L^\Phi (\Omega )$ if and only if the measure of singular points of $\Phi$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $\Phi$, which implies that the measure of the singular points of $\Phi$ is zero. As a corollary we obtain analogous results for Musielak–Orlicz spaces generated by double phase functional and we recover the well-known result for variable exponent Lebesgue spaces.