Morrey-Sobolev Spaces on Metric Measure Spaces
Yufeng Lu, Dachun Yang, Wen Yuan

TL;DR
This paper introduces and studies Morrey-Sobolev spaces on metric measure spaces, establishing embedding theorems, equivalences with Haj{ extl}asz-based spaces, and boundedness of maximal operators, extending classical analysis to more general spaces.
Contribution
The paper defines Newton-Morrey-Sobolev and Haj{ extl}asz-Morrey-Sobolev spaces on metric measure spaces and proves their equivalence under certain conditions, extending Sobolev space theory.
Findings
Embedding into H"older spaces under Poincaré and doubling conditions
Equivalence of Newton-Morrey-Sobolev and Haj{ extl}asz-Morrey-Sobolev spaces
Boundedness of fractional maximal operators on Morrey-type spaces
Abstract
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space . The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if supports a weak Poincar\'e inequality and the measure is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors -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 is doubling and supports a weak Poincar\'e inequality. In particular, on the Euclidean space , the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj{\l}asz-Morrey-Sobolev space and the classical…
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Nonlinear Partial Differential Equations · Geometric Analysis and Curvature Flows
