Scalefree unique continuation estimates and LogvinenkoSereda 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 timedependent 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 timedependent 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.07020
 Bibcode:
 2016arXiv160907020E
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs