Unique continuation estimates for Baouendi--Grushin equations on cylinders

TL;DR

This paper establishes quantitative unique continuation estimates and spectral inequalities for fractional Baouendi--Grushin operators on cylinders, advancing understanding of controllability and spectral properties of these degenerate operators.

Contribution

It provides new time-pointwise estimates and spectral inequalities for fractional Baouendi--Grushin operators on cylinders, extending previous results in the field.

Findings

Quantitative unique continuation estimates established.

Spectral inequalities relating norms on the whole cylinder and subsets derived.

Results on exact and approximate null-controllability obtained.

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 $\mathbb{R}^d \times \mathbb{T}^d$. 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.