Variations on Liouville's theorem

A classical theorem of Liouville states that a function that is analytic and bounded on the entire complex plane is in fact constant. The same conclusion is true for a function that is harmonic and bounded on $\mathbb{R}^n$.

The talk discusses generalisations of Liouville's theorem to nonlocal translation-invariant operators. It is based on a joint work with D. Berger and R.L. Schilling, and a further joint work with the same co-authors and T. Sharia. We consider operators with continuous but not necessarily infinitely smooth symbols.

It follows from our results that if $\left\{\eta \in \mathbb{R}^n \mid  m(\eta) = 0\right\} \subseteq \{0\}$, then, under suitable conditions, every polynomially bounded weak solution $f$ of the equation $m(D)f=0$ is in fact a polynomial, while sub-exponentially growing solutions admit analytic continuation to entire functions on $\mathbb{C}^n$.