THE CRAMER-WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS

Karlheinz Gröchenig (1) , Philippe Jaming (2)

1 Fakultät für Mathematik [Wien]
2 IMB - Institut de Mathématiques de Bordeaux

Two measurable sets S, Λ ⊆ R d form a Heisenberg uniqueness pair, if every bounded measure µ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in R d. As a corollary we obtain a new, surprising version of the classical Cramér-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 .

Mots clés: Heisenberg Uniqueness; Cramer-Wold theorem; Unique continuation

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Karlheinz Gröchenig, Philippe Jaming. THE CRAMER-WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS. Journal de l'Institut de Mathématiques de Jussieu, 2020, 19, pp.117-135. ⟨hal-01355577⟩