Liouville's theorems for L\'evy operators

Let $L$ be a L\'evy 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 L\'evy 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 L\'evy operator $L$ does not hold.Comment: 43 page