Let L be a Lévy operator: a translation-invariant non-local “elliptic” operator. A function h is said to be harmonic with respect to L if Lh = 0 in an appropriate sense. A Liouville's theorem for an operator L is a result which, under appropriate assumptions, characterises the class of functions harmonic with respect to L. The most common variant states that bounded harmonic functions are necessarily constants. For general Lévy operators this was proved by Alibaud, del Teso, Endal and Jakobsen in 2020, and independently by Berger and Schilling in 2022. In a joint work with Tomasz Grzywny we extend the above results in three directions.
First, we prove that positive harmonic functions are mixtures of harmonic exponentials, in the spirit of Deny's theorem for convolution equations and a less general result for Lévy operators given by Berger and Schilling.
Next, we provide a variant of Liouville's theorem for signed harmonic functions, which is similar to the result obtained independently by Berger, Schilling and Shargorodsky in 2024, and which asserts that, under appropriate assumptions, signed harmonic functions are necessarily harmonic polynomials. Although not completely general, our result extends all previously known theorems of this kind.
Finally, we provide an explicit example of a Lévy operator L and a signed, polynomially bounded function h which show that without any conditions, Liouville's theorem fails: h is harmonic with respect to L, but not a polynomial.