Revisit on Heisenberg uniqueness pair for the hyperbola

Abstract

Let $\Gamma$ be the hyperbola $\{(x,y)\in \mathbb {R}^2:xy=1\}$ and $\Lambda _{\alpha , \beta ,\theta _1, \theta _2}$ be the perturbed lattice-cross defined by $\Lambda _{\alpha , \beta , \theta _1, \theta _2}=\left( (\alpha \mathbb Z+\{\theta _1\})\times \{0\}\right) \cup \left( \{0\}\times (\beta \mathbb Z+\{\theta _2\})\right)$ in $\mathbb {R}^2$, where $\theta _1, \theta _2\in \mathbb R,$ and $\alpha , \beta$ are positive reals. Under certain conditions on the parameters $\alpha , \beta ,\theta _1$ and $\theta _2$, we study necessary and sufficient conditions for Heisenberg uniqueness pairs corresponding to the hyperbola. Our method of proof is inspired by the work of Hedenmalm and Montes-Rodríguez where they considered the classical case, that is, $\theta _1=\theta _2=0$. Moreover, we answer an interesting question raised by Canto-Martín, Hedenmalm, and Montes-Rodríguez, related to an explicit formulation of the certain pre-annihilator space.