Revisit on Heisenberg uniqueness pair for the hyperbola

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2024年2月8日

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Debkumar Giri,
Ramesh Manna

摘要

Let $\Gamma $ $Γ$ be the hyperbola $\{(x,y)\in \mathbb {R}^2:xy=1\}$ ${(x, y) \in R^{2} : x y = 1}$ and $\Lambda _{\alpha , \beta ,\theta _1, \theta _2}$ $Λ_{α, β, θ_{1}, θ_{2}}$ 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) $ $Λ_{α, β, θ_{1}, θ_{2}} = ((α Z + {θ_{1}}) \times {0}) \cup ({0} \times (β Z + {θ_{2}}))$ in $\mathbb {R}^2$ $R^{2}$ , where $\theta _1, \theta _2\in \mathbb R,$ $θ_{1}, θ_{2} \in R,$ and $\alpha , \beta $ $α, β$ are positive reals. Under certain conditions on the parameters $\alpha , \beta ,\theta _1$ $α, β, θ_{1}$ and $\theta _2$ $θ_{2}$ , 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, $\theta _1=\theta _2=0$ $θ_{1} = θ_{2} = 0$ . 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.