Abstract

We prove time-pointwise quantitative unique continuation estimates for the evolution operators associated to (fractional powers of) the Baouendi–Grushin operators on the cylinder ${ℝ}^d\times {\mathbb{𝕋}}^d$. Spectral inequalities are deduced, relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole cylinder ${ℝ}^d\times {\mathbb{𝕋}}^d$ to the $L^2$-norm on so-called thick subsets of ${ℝ}^d\times {\mathbb{𝕋}}^d$. As a byproduct, we also obtain results on exact and approximate null-controllability for the corresponding evolution systems. This extends and complements results obtained recently by the authors and by Jaming and Wang.

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