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

A pair $(Γ,Λ)$, where $Γ\subset\mathbb{R}^2$ is a locally rectifiable curve and $Λ\subset\mathbb{R}^2$ is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on $Γ$ whose Fourier transform vanishes on $Λ$ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if $Γ$ is the hyperbola $x_1x_2=M^2/(4π^2)$, where $M>0$ is the mass, and $Λ$ is the lattice-cross $(α\mathbb{Z}\times\{0\}) \cup (\{0\}\timesβ\mathbb{Z})$, where $α,β$ are positive reals, then $(Γ,Λ)$ is a Heisenberg uniqueness pair if and only if $αβM^2\le4π^2$. 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 $0<αβ<4$, 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 $0<αβ\le1$. 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.

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