Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus

Abstract

We study uncertainty principles for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier transform of the considered functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the density and scale of the observability set. Our results are on the one hand closely related to unique continuation for linear combinations of eigenfunctions (aka spectral inequalities) which can be obtained by Carleman estimates, on the other hand to observability estimates for the time-dependent Schroedinger and for the heat equation, and finally to the Logvinenko & Sereda theorem. In fact, they are based on the methods developed by Kovrijkine to refine and generalize the results of Logvinenko & Sereda and Kacnel'son. Furthermore, relying on completely different techniques associated with the time-dependent Schroedinger equation, we prove a companion theorem where the energy of the considered functions is allowed to be in a spectral subspace of a Schroedinger operator.

28 pages. New material added. Accepted for publication in Annales Henri Poincaré

References in corpus (5)

• Nazarov's uncertainty principles in higher dimension
• Sharp geometric condition for null-controllability of the heat equation on ${\mathbb{R}}^d$ and consistent estimates on the control cost
• Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators
• Sharp estimates and homogenization of the control cost of the heat equation on large domains
• Sufficient criteria and sharp geometric conditions for observability in Banach spaces

Cited by in corpus (9)

• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Sharp estimates and homogenization of the control cost of the heat equation on large domains
• Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
• Uncertainty principles with error term in Gelfand-Shilov spaces
• Unique continuation estimates for Baouendi--Grushin equations on cylinders
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs