Density of smooth functions in Musielak-Orlicz spaces

TL;DR

This paper establishes necessary and sufficient conditions for the density of smooth functions in Musielak-Orlicz spaces, extending previous results and including special cases like variable exponent Lebesgue spaces.

Contribution

It provides a complete characterization of when smooth functions are dense in Musielak-Orlicz spaces, especially under the $ riangle_2$ condition, generalizing earlier theorems.

Findings

Density of smooth functions characterized by measure of singular points

Extension of density results to spaces generated by double phase functional

Recovery of classical results for variable exponent Lebesgue 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.