Unique continuation estimates for Baouendi--Grushin equations on cylinders

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