Functional spaces of two variable exponents on metric measure spaces

我们的目的是研究度量空间上的Musielak-Orlicz Sobolev空间。我们考虑Hajlasz型条件和牛顿型条件。证明了Lipschitz连续函数是稠密的，并证明了它的一些基本性质。我们研究了这些空间之间的关系，并讨论了这些空间中的勒贝格点定理。我们还讨论了Hardy-Littlewood极大算子在Musielak-Orlicz空间上的有界性。作为Hardy-Littlewood极大算子有界性的一个应用，我们在Musielak-Orlicz-Hajlasz空间中建立了Soblev函数的Soblev不等式的一个推广。

Our aim is to study Musielak-Orlicz Sobolev spaces on metric measure spaces. We consider a Hajlasz type condition and a Newtonian type condition. We prove that Lipschitz continuous functions are dense as well as other basic properties. We study the relationship between these spaces, and discuss Lebesgue point theorem in these spaces. We also deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz spaces. As an application of the boundedness of the Hardy-Littlewood maximal operator, we establish a generalization of Sobolev's inequality for Sobolev functions in Musielak-Orlicz-Hajlasz spaces.