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.
