Harmonic functions on 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-Savaré. 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.

21 pages, comments are welcome!

References in corpus (6)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• The splitting theorem in non-smooth context
• Li-Yau and Harnack type inequalities in $RC{D}^{\ast }(K,N)$ metric measure spaces
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Cited by in corpus (9)

• On Interpolation and Curvature via Wasserstein Geodesics
• Hamilton's Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces
• Weyl's law on $RC{D}^{\ast }(K,N)$ metric measure spaces
• $(K,N)$-convexity and the curvature-dimension condition for negative $N$
• Cheeger-Colding-Tian theory for conic Kahler-Einstein metrics
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces
• Monotonicity formulas for harmonic functions in $\mathrm{RCD}(0,N)$ spaces
• On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth
• Mean value property and harmonicity on Carnot-Carathéodory groups