Density of smooth functions in Musielak–Orlicz spaces

Abstract

We provide necessary and sufficient conditions for the space of smooth functions with compact supports Cc∞(Ω) to be dense in Musielak–Orlicz spaces LΦ(Ω) where Ω is an open subset of Rd. In particular, we prove that if Φ satisfies condition Δ 2, the closure of Cc∞(Ω)∩LΦ(Ω) is equal to LΦ(Ω) if and only if the measure of singular points of Φ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of Φ , which implies that the measure of the singular points of Φ 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.

Publication Title

Banach Journal of Mathematical Analysis

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