APDE seminar@BCAM: Uniqueness and non-uniqueness pairs for the fractional Laplacian

Abstract

A uniqueness pair problem asks whether a function is uniquely determined by prescribing its values on one set and the values of a given operator applied to it on another set. In this talk, we examine this question for discrete sets in the context of the fractional Laplacian. More precisely, let $\Lambda, M \subset \mathbb{R}^d$ be discrete sets, and let $f$ be a sufficiently regular function that vanishes on $\Lambda$, and suppose that its fractional Laplacian $(-\Delta)^s f$ vanishes on $M$. We give sufficient conditions on the sets $\Lambda$ and $M$ under which $f$ is identically zero. Our approach is inspired by earlier work of Ramos and Sousa, as well as Kulikov, Nazarov, and Sodin. We also show that these methods extend naturally to a broader class of multiplier operators.