Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of H\"ormander's operators

TL;DR

This paper establishes the precise domain of dependence for subelliptic wave equations and proves unique continuation properties for fractional powers of H"ormander's operators, advancing understanding of these operators' behavior.

Contribution

It introduces sharp domain of dependence results and extends unique continuation properties to fractional powers of subelliptic operators under analyticity conditions.

Findings

Proved sharp domain of dependence for subelliptic wave equations.

Established unique continuation for square roots of subelliptic Laplacians.

Extended unique continuation results to fractional powers between 0 and 1.

Abstract

We prove the sharp domain of dependence property for solutions to subelliptic wave equations for sums of squares of vector fields satisfying H\"ormander bracket condition. We deduce a unique continuation property for the square root of subelliptic Laplace operators under an additional analyticity condition. Then, with a different, more involved method, we prove the same result of unique continuation for more general $s$-powers ($0<s<1$).