论文标题
Musielak-Orlicz空间中平滑功能的密度
Density of smooth functions in Musielak-Orlicz spaces
论文作者
论文摘要
我们提供了带有紧凑型功能的平滑功能空间的必要条件。特别是我们证明，如果$φ$满足条件$δ_2$，则$ c^\ infty_c（ω）\ cap l^φ（ω）$的关闭等于$ l^φ（ω）$，并且仅当$φ$的单数点的度量等于$ $ $等于零。这扩展了在$φ$的局部集成性下证明的早期密度定理，这意味着$φ$的单数点的度量为零。作为推论，我们获得了由双相功能产生的Musielak-orlicz空间的类似结果，并为可变指数Lebesgue空间恢复了已知的结果。
We provide necessary and sufficient conditions for the space of smooth functions with compact supports $C^\infty_C(Ω)$ to be dense in Musielak-Orlicz spaces $L^Φ(Ω)$ where $Ω$ is an open subset of $\mathbb{R}^d$. In particular we prove that if $Φ$ satisfies condition $Δ_2$, the closure of $C^\infty_C(Ω)\cap L^Φ(Ω)$ 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.
