IS06: Lévy-Type Processes

Organizer: Victoria Knopova (Kyiv Taras Shevchenko National University)

Liouville theorems for Fourier Multipliers

David Berger

In this talk we will discuss Liouville theorems for a class of Fourier multipliers, which contains especially the generators of Lévy operators. We are looking at the bounded case and show that with some adjustments the proof goes through for functions satisfying certain growth conditions. In particular, we present new results for Fourier Mulipliers in dimension \(1\). At the end we present new results for the unique continuation properties of non-local operators.

Bibliography

\([1]\) D. Berger, R. Schilling. The (strong) Liouville property for a class of non-local operators, Math. Scand. (2022), 128(2).

\([2]\) D. Berger, R. Schilling, E. Shargordosky The Liouville Theorem for a class of Fourier Multipliers and its connection to coupling. B. Lond. Math. Soc., appeared online first, DOI: 10.1112/blms.1306 (2024).

\([3]\) D. Berger, R. Schilling, E. Shargordosky, T. Sharia. An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions. J. Spectr. Theory (2024), 14(2).

\([4]\) D. Berger, R. Schilling. On the unique continuation property of non-local operators. Working paper.