Morrey-Sobolev Spaces on Metric Measure Spaces

Abstract

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathcal{X},d,\mu )$. The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if $\mathcal{X}$ supports a weak Poincaré 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 Hajłasz gradient, the authors also introduce the Hajłasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajłasz-Morrey-Sobolev space when $\mu$ is doubling and $\mathcal{X}$ supports a weak Poincaré inequality. In particular, on the Euclidean space ${\mathbb{R}}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Hajłasz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\mathcal{X},d)$ 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 Hajłasz-Morrey-Sobolev spaces.

Potential Anal. (2013), DOI: 10.1007/s11118-013-9370-9

References in corpus (3)

• Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices
• Fractional maximal functions in metric measure spaces
• Hajlasz Gradients Are Upper Gradients