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L q harmonic functions on graphs

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Abstract

We prove an analogue of Yau’s Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or nonnegative subharmonic functions of class L q, 1 ≤ q < ∞, on any graph, and a quantitative version for q > 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.

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Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Bobo Hua & Jürgen Jost

  2. Department of Mathematics and Computer Science, University of Leipzig, 04109, Leipzig, Germany

    Jürgen Jost

Authors
  1. Bobo Hua
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  2. Jürgen Jost
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Corresponding author

Correspondence to Bobo Hua.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 267087.

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Hua, B., Jost, J. L q harmonic functions on graphs. Isr. J. Math. 202, 475–490 (2014). https://doi.org/10.1007/s11856-014-1089-9

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  • DOI: https://doi.org/10.1007/s11856-014-1089-9

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