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.