The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

TL;DR

This paper establishes conditions under which measures supported on quadratic surfaces are uniquely determined by their Fourier transforms on certain sets, extending classical results and applying to PDE eigenfunctions.

Contribution

It proves that quadratic hypersurfaces and unions of hyperplanes form Heisenberg uniqueness pairs, leading to a new version of the Cramér-Wold theorem for measures on quadratic surfaces.

Findings

Quadratic hypersurfaces and two hyperplanes form Heisenberg uniqueness pairs.

Measures on quadratic surfaces are determined by projections onto two hyperplanes.

Application to unique continuation of PDE eigenfunctions.

Abstract

Two measurable sets $S, \Lambda \subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $\mu$ with support in S whose Fourier transform vanishes on {\Lambda} must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cram\'er-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients .