An abstract Logvinenko-Sereda type theorem for spectral subspaces

TL;DR

This paper extends the Logvinenko-Sereda theorem to spectral subspaces of self-adjoint operators, providing explicit bounds for functions based on their behavior on thick subsets, thus broadening classical results to new domains.

Contribution

It introduces an abstract framework replacing Fourier support with spectral subspaces, enabling the extension of classical inequalities to more general settings.

Findings

Provides explicit $L^2$-norm bounds in spectral subspaces.

Recovers classical results for Fourier support and Hermite functions.

Extends inequalities to new domains using Bernstein-type inequalities.

Abstract

We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean $L^2$-space. Our result then provides a bound for the $L^2$-norm of such functions in terms of their $L^2$-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernstein-type inequalities for the respective functions, complemented with a suitable covering of the underlying domain.