Sharp resolvent estimate for the Baouendi-Grushin operator and applications

TL;DR

This paper establishes sharp resolvent estimates for the Baouendi-Grushin operator on the torus with H"older damping, revealing how damping geometry influences energy decay rates in subelliptic and elliptic regimes.

Contribution

It provides the first sharp resolvent bounds for the non-selfadjoint Baouendi-Grushin operator under various damping configurations, using advanced microlocal analysis techniques.

Findings

Sharp resolvent estimates are obtained in all damping scenarios.

Optimal energy decay rates are established for the damped wave equations.

The results contrast with classical Laplace resolvent behavior in subelliptic regimes.

Abstract

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.