Abstract
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima’s ergodic theorem for the harmonic functions in the domain of the \( L^{p} \) generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain \( L^{p} \) growth criteria must be constant. As consequence we give an integral criterion for recurrence.
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The last three authors acknowledge the financial support of the German Science Foundation (DFG).
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Hua, B., Keller, M., Lenz, D., Schmidt, M. (2022). On \(L^p\) Liouville Theorems for Dirichlet Forms. In: Chen, ZQ., Takeda, M., Uemura, T. (eds) Dirichlet Forms and Related Topics. IWDFRT 2022. Springer Proceedings in Mathematics & Statistics, vol 394. Springer, Singapore. https://doi.org/10.1007/978-981-19-4672-1_12
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DOI: https://doi.org/10.1007/978-981-19-4672-1_12
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