Bobo Hua, Matthias Keller
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, various quantitative results for graphs of finite measure are obtained.
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http://arxiv.org/abs/1303.7198
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