Square roots of perturbed subelliptic operators on Lie groups

TL;DR

This paper addresses the Kato square root problem for perturbed subelliptic operators on Lie groups, establishing inhomogeneous and homogeneous estimates, and demonstrating stability under small coefficient perturbations.

Contribution

It extends the Kato square root problem to subelliptic operators on Lie groups with measurable perturbations, including stability results.

Findings

Established inhomogeneous estimates for perturbed subelliptic operators.

Proved stronger homogeneous estimates for nilpotent groups with second order operators.

Demonstrated Lipschitz stability of estimates under small coefficient perturbations.

Abstract

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, then we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small perturbations of the coefficients.