Quantitative spectral inequalities for the anisotropic Shubin operators and the Grushin operator
Résumé
We prove quantitative spectral inequalities for the (anisotropic) Shubin operators and the Grushin operator on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole space to the $L^2$-norm on a suitable subset. A particular feature of our estimates is that the constant relating these $L^2$-norms is very explicit in geometric parameters of the corresponding subset of the whole space. While these subsets may be thick (i.e. relatively dense) in the classical sense in the case of the Grushin operator, they may become sparse at infinity for the Shubin operators and may even have finite measure. The latter extends results obtained recently by J. Martin and, in the particular case of the harmonic oscillator, by A. Dicke, I. Veselić, and the second author. We apply our results towards null-controllability of the associated (degenerate) parabolic equations, as well as to the ones associated to the Baouendi-Grushin operators acting on $\mathbb R^d\times\mathbb T^d$.
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