Morrey-Sobolev Spaces on Metric Measure Spaces

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak Poincar\'e inequality and the measure $\mu$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj{\l}asz gradient, the authors also introduce the Haj{\l}asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj{\l}asz-Morrey-Sobolev space when $\mu$ is doubling and $\mathscr{X}$ supports a weak Poincar\'e inequality. In particular, on the Euclidean space ${\mathbb R}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj{\l}asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\mathscr{X},d)$ is geometrically doubling and $\mu$ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when $\mu$ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj{\l}asz-Morrey-Sobolev spaces.