Morrey-Sobolev Spaces on Metric Measure Spaces

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space (oe'(3), d, mu). The embedding of the Newton-Morrey-Sobolev space into the Holder space is obtained if oe'(3) supports a weak Poincar, 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 Hajasz gradient, the authors also introduce the Hajasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajasz-Morrey-Sobolev space when mu is doubling and oe'(3) supports a weak Poincar, inequality. In particular, on the Euclidean space , the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Hajasz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when (oe'(3), 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 Hajasz-Morrey-Sobolev spaces.