The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps

TL;DR

This paper investigates the uniqueness properties of solutions to the Klein-Gordon equation supported on hyperbolas, using harmonic analysis and dynamics of Gauss-type maps to characterize when certain exponential systems are dense.

Contribution

It completely solves the problem of characterizing weak-star density of exponential systems on semi-infinite axes and holomorphic spaces, extending previous results on Heisenberg uniqueness pairs.

Findings

Weak-star density on $ ext{R}_+$ holds if and only if $0<eta ext{α}<4$.

Weak-star density in $H^ ext{infty}_+( ext{R})$ holds if and only if $0<eta ext{α} ext{≤}1$.

Develops new harmonic analysis tools related to Gauss-type maps and the Hilbert transform.

Abstract

A pair $(\Gamma,\Lambda)$, where $\Gamma\subset\mathbb{R}^2$ is a locally rectifiable curve and $\Lambda\subset\mathbb{R}^2$ is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on $\Gamma$ whose Fourier transform vanishes on $\Lambda$ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if $\Gamma$ is the hyperbola $x_1x_2=M^2/(4\pi^2)$, where $M>0$ is the mass, and $\Lambda$ is the lattice-cross $(\alpha\mathbb{Z}\times\{0\}) \cup (\{0\}\times\beta\mathbb{Z})$, where $\alpha,\beta$ are positive reals, then $(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair if and only if $\alpha\beta M^2\le4\pi^2$. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. The case of the semi-axis $\mathbb{R}_+$ as well as the holomorphic counterpart remained open. In this work, we completely solve these two problems. As for the semi-axis, we show that the restriction to $\mathbb{R}_+$ of the above exponential system spans a weak-star dense subspace of $L^\infty(\mathbb{R}_+)$ if and only if $0<\alpha\beta<4$, based on dynamics of Gauss-type maps. This has an interpretation in terms of dynamical unique continuation. As for the holomorphic counterpart, we show that the above exponential system with $m,n\ge0$ spans a weak-star dense subspace of $H^\infty_+(\mathbb{R})$ if and only if $0<\alpha\beta\le1$. To obtain this result, we need to develop new harmonic analysis tools for the dynamics of Gauss-type maps, related to the Hilbert transform. Some details are deferred to a separate publication.