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Explore advanced mathematical analysis in this 56-minute lecture examining Liouville's theorems for Lévy operators, delivered as part of the Erwin Schrödinger International Institute's Thematic Programme on Free Boundary Problems. Delve into the theoretical foundations of Lévy operators, which are translation-invariant non-local "elliptic" operators, and understand the concept of harmonic functions where Lh = 0 in an appropriate mathematical sense. Learn about the classical Liouville theorem stating that bounded harmonic functions are necessarily constants, as proven for general Lévy operators by Alibaud, del Teso, Endal and Jakobsen in 2020, and independently by Berger and Schilling in 2022. Discover three significant extensions to these foundational results through joint research with Tomasz Grzywny. First, examine how positive harmonic functions can be characterized as mixtures of harmonic exponentials, building upon Deny's theorem for convolution equations and extending previous work by Berger and Schilling. Second, investigate a variant of Liouville's theorem for signed harmonic functions, demonstrating that under appropriate assumptions, these functions are necessarily harmonic polynomials, similar to independent results by Berger, Schilling and Shargorodsky in 2024. Finally, analyze an explicit counterexample featuring a Lévy operator L and a signed, polynomially bounded function h that demonstrates the failure of Liouville's theorem without proper conditions, where h remains harmonic with respect to L but is not polynomial in nature.
