"Density of smooth functions in Musielak–Orlicz spaces" by Anna Kamińska and Mariusz Żyluk
 

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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