## Abstract

An improvement of the Liouville theorem for discrete harmonic functions on Z^{2} is obtained. More precisely, we prove that there exists a positive constant " such that if u is discrete harmonic on Z^{2} and for each sufficiently large square Q centered at the origin \ u\ ≤ 1 on a (1 − ε ) portion of Q, then u is constant.

Original language | English |
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Pages (from-to) | 1349-1378 |

Number of pages | 30 |

Journal | Duke Mathematical Journal |

Volume | 171 |

Issue number | 6 |

DOIs | |

State | Published - 15 Apr 2022 |

### Funding

Funders | Funder number |
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European Commission | 382/15, 213638, 692616 |

Israel Science Foundation | 1380/13 |

Norges Forskningsråd |

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