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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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Acknowledgments

BH thanks Jürgen Jost for inspiring discussions on \(L^p\) Liouville theorem and constant support, and acknowledges the financial support from the funding of the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 267087. MK enjoyed discussions with Gabor Lippner, Dan Mangoubi, Marcel Schmidt and Radosław Wojciechowski on the subject and acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no. 2010214.

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Communicated by J. Jost.

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Hua, B., Keller, M. Harmonic functions of general graph Laplacians. Calc. Var. 51, 343–362 (2014). https://doi.org/10.1007/s00526-013-0677-6

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