Heisenberg uniqueness pairs
Speaker 
Dr. Somnath Ghosh,
IIT, Guwahati


When 
Feb 09, 2021
from 03:00 PM to 04:00 PM 
Where  zoom meet 
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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 MontesRodriguez in 2011.
Suppose \(X(\Gamma)\) be the space of all finite complexvalued 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 wellknown 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.