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In this article, the authors introduce the Newton-Morrey-Sobolev space on ametric measure space $(\mathscr{X},d,\mu)$. The embedding of theNewton-Morrey-Sobolev space into the H\"older space is obtained if$\mathscr{X}$ supports a weak Poincar\'e inequality and the measure $\mu$ isdoubling 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 theHaj{\l}asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolevspace coincides with the Haj{\l}asz-Morrey-Sobolev space when $\mu$ is doublingand $\mathscr{X}$ supports a weak Poincar\'e inequality. In particular, on theEuclidean space ${\mathbb R}^n$, the authors obtain the coincidence among theNewton-Morrey-Sobolev space, the Haj{\l}asz-Morrey-Sobolev space and theclassical Morrey-Sobolev space. Finally, when $(\mathscr{X},d)$ isgeometrically doubling and $\mu$ a non-negative Radon measure, the boundednessof some modified (fractional) maximal operators on modified Morrey spaces ispresented; as an application, when $\mu$ is doubling and satisfies some measuredecay property, the authors further obtain the boundedness of some (fractional)maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces andHaj{\l}asz-Morrey-Sobolev spaces.Comment: Potential Anal. (2013), DOI: 10.1007/s11118-013-9370-

Morrey-Sobolev Spaces on Metric Measure Spaces