2013/04/14 by Yufeng Lu, Dachun Yang, Lu, Yufeng +3
Mathematics · #30L99 #42B35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary 46E35 #Secondary 42B25
paper · pdf · doi:10.48550/arxiv.1304.3871
openalex publication_date 2013/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space (\mathscrX,d,μ). The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if \mathscrX supports a weak Poincaré inequality and the measure μ 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 μ is doubling and \mathscrX supports a weak Poincaré inequality. In particular, on the Euclidean space \mathbb Rn, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Hajłasz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when (\mathscrX,d) is geometrically doubling and μ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when μ 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.