Carleman estimates for sub-Laplacians on Carnot groups

TL;DR

This paper develops a new Carleman estimate with singular weights for sub-Laplacians on Carnot groups, enabling sharp vanishing order estimates for solutions to stationary Schrödinger equations, advancing analysis in sub-Riemannian geometry.

Contribution

It introduces a novel Carleman estimate with singular weights for sub-Laplacians on Carnot groups, leading to improved vanishing order bounds for Schrödinger solutions.

Findings

Established a new Carleman estimate with singular weights.

Derived sharp vanishing order estimates for Schrödinger solutions.

Applicable to functions satisfying a specific discrepancy assumption.

Abstract

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group $\mathbb G$ for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schr\"odinger equations.