A DISCRETE HARMONIC FUNCTION BOUNDED ON A LARGE PORTION OF Z2 IS CONSTANT

Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova, Mikhail Sodin

Research output: Contribution to journalArticlepeer-review

Abstract

An improvement of the Liouville theorem for discrete harmonic functions on Z2 is obtained. More precisely, we prove that there exists a positive constant " such that if u is discrete harmonic on Z2 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 languageEnglish
Pages (from-to)1349-1378
Number of pages30
JournalDuke Mathematical Journal
Volume171
Issue number6
DOIs
StatePublished - 15 Apr 2022

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