An abstract Logvinenko-Sereda type theorem for spectral subspaces

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.

arXiv comment31 pages, 2 figures; minor changes and fixed typos, added references, and updated bibliography

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Cited by in corpus (11)

• Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
• The reflection principle in the control problem of the heat equation
• 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
• The Laplacian on Cartesian products with mixed boundary conditions
• Uncertainty principles with error term in Gelfand-Shilov spaces
• Spectral estimate for the Laplace-Beltrami operator on the hyperbolic half-plane
• Unique continuation estimates for Baouendi--Grushin equations on cylinders
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Null-controllability of the Generalized Baouendi-Grushin heat like equations
• Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs