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

Abstract

We study the uncertainty principle associated with the Klein–Gordon equation. As in the previous work [Ann. of Math. 173 (2011)], we consider vanishing along a lattice-cross. The following variants appear naturally: (1) vanishing only along “half” of the lattice-cross, where the “half” is defined as being on the boundary of a quarter-plane, and (2) that the function vanishes on the whole lattice-cross, but we require the function to have Fourier transform supported by one of the two branches of the hyperbola. In case (1) the critical phenomenon is whether the given condition forces the function to vanish on the quarter-plane in question. Here it turns out to be crucial whether the quarter-plane is space-like or time-like, and in short the answer is yes for spacelike and no for time-like. The analysis brings us quite far, involving the orbit of the Hilbert kernel under the iterates of the transfer operator, and uses methods from the theory of totally positive matrices as well as Hurwitz zeta functions, and is partially postponed to a separate publication. In case (2), the critical phenomenon occurs at another density, and the dynamics then comes from the standard Gauss transformation t 7→ 1/t mod Z on the interval [0, 1]. In the intermediate range of the density of the lattice-cross, we obtain unique extendability of the Fourier transform from one branch of the hyperbola to the other.