Density of smooth functions in Musielak-Orlicz spaces

2022/01/14 by Kamińska, Anna, Żyluk, Mariusz
#46B42 #46E15 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2201.05694

Abstract

We provide necessary and sufficient conditions for the space of smooth functions with compact supports C^∞_C(Ω) to be dense in Musielak-Orlicz spaces L^Φ(Ω) where Ω is an open subset of ℝ^d. In particular we prove that if Φ satisfies condition Δ₂, the closure of C^∞_C(Ω)∩ L^Φ(Ω) is equal to L^Φ(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.

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