Heisenberg uniqueness pairs

Abstract: In this talk, we shall consider a recent variant of the uncertainty principle for the Fourier transform, namely, Heisenberg uniqueness pair (HUP). The concept of HUP was first introduced by Hedenmalm and Montes-Rodriguez in 2011.

Suppose \(X(\Gamma)\) be the space of all finite complex-valued Borel measures \(\mu\) in \(\mathbb R^2,\) which are supported on \(\Gamma,\) finite disjoint union of smooth curves, and absolutely continuous with respect to the arc length measure of \(\Gamma.\) For a set \(\Lambda\) in \(\mathbb R^2,\) the pair \(\left(\Gamma, \Lambda\right)\) is called a HUP for \(X(\Gamma)\) if the only \(\mu\in X(\Gamma)\) whose Fourier transform satisfies \(\hat\mu\vert_\Lambda=0\) is \(\mu=0.\) First, we shall start with some HUPs for measures supported on well-known curves in the Euclidean spaces and discuss how it correlates with the uniqueness properties of solutions of certain PDEs. Then we shall extend the concept of HUP on the Heisenberg group, and for measures supported on the sphere, we talk about some determining sets.