Harmonic functions of general graph Laplacians |
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Authors: | Bobo Hua Matthias Keller |
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Institution: | 1. Max Planck Institute for Mathematics in the Sciences, 04103?, Leipzig, Germany 2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904?, Jerusalem, Israel
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Abstract: | We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an \(L^{p}\) Liouville type theorem which is a quantitative integral \(L^{p}\) estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s \(L^{p}\) -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on \(L^{p}\) and get a criterion for recurrence. As a side product, we show an analogue of Yau’s \(L^{p}\) Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces. |
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