Harmonic functions on metric measure spaces
Bobo Hua, Martin Kell, Chao Xia
TL;DR
This paper investigates harmonic functions on metric measure spaces with lower Ricci curvature bounds, establishing gradient estimates and dimension bounds for polynomial growth harmonic functions in this setting.
Contribution
It provides the first Cheng-Yau type gradient estimate and optimal dimension bounds for harmonic functions on such metric measure spaces.
Findings
Proved a local gradient estimate for harmonic functions.
Derived optimal dimension estimates for polynomial growth harmonic functions.
Extended classical results to non-smooth metric measure spaces.
Abstract
In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds