Heisenberg uniqueness pairs and the wave equation

Abstract

The concept of the Heisenberg uniqueness pair $(\Gamma ,\Lambda )$ was introduced by Hedenmalm and Montes-Rodríguez as a variant of the uncertainty principle for the Fourier transform. The main results in Hedenmalm and Montes-Rodríguez (Ann. of Math. (2) 173 (2011), 1507–1527) concern the hyperbola ${\Gamma }_{\epsilon }=\{(x_1,x_2)\in R^2,x_1{x}_2=\epsilon \}$ ($0\ne \epsilon \in R$) and lattice-crosses ${\Lambda }_{\alpha \beta }=(\alpha Z\times \left\{0\right\})\cup (\left\{0\right\}\times \beta Z)$ ($\alpha ,\beta >0$), where it is proved that $({\Gamma }_{\epsilon },{\Lambda }_{\alpha \beta })$ is a Heisenberg uniqueness pair if and only if $\alpha \beta ⩽1/\left|\epsilon \right|$. In this paper, we study the endpoint case ($\epsilon =0$ in ${\Gamma }_{\epsilon }$) and investigate the minimal information required on the zero set $\Lambda$ to form such a pair. When $\Lambda$ is contained in the union of two curves in the plane, we provide characterizations using dynamical system conditions. The situation differs in higher dimensions, where we obtain characterizations when $\Lambda$ is the union of two hyperplanes.