Liouville's theorems for Lévy operators

2023/01/20 by Tomasz Grzywny, Grzywny, Tomasz, Mateusz Kwaśnicki +1 · 2 citations
Engineering · Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.08540

openalex publication_date 2023/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a Lévy operator. A function h is said to be harmonic with respect to L if L h = 0 in an appropriate sense. We prove Liouville's theorem for positive functions harmonic with respect to a general Lévy operator L: such functions are necessarily mixtures of exponentials. For signed harmonic functions we provide a fairly general result, which encompasses and extends all Liouville-type theorems previously known in this context, and which allows to trade regularity assumptions on L for growth restrictions on h. Finally, we construct an explicit counterexample which shows that Liouville's theorem for signed functions harmonic with respect to a general Lévy operator L does not hold.

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