Propagation of singularities for subelliptic wave equations

TL;DR

This paper extends the understanding of singularity propagation in subelliptic wave equations, showing they follow null-bicharacteristics and abnormal extremals, which impacts the link between sub-Riemannian geometry and sub-Laplacians.

Contribution

It proves that singularities in subelliptic wave equations propagate along specific curves, including abnormal extremals, refining previous theories and characterizing the singular support of wave kernels.

Findings

Singularities propagate along null-bicharacteristics and abnormal extremals.

Characterization of the singular support of subelliptic wave kernels outside the diagonal.

Abnormal extremals are crucial in the classical-quantum correspondence in sub-Riemannian geometry.

Abstract

H{\"o}rmander's propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a visionary conference paper by R. Melrose \cite{Mel86}, we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremals, which are well-known curves in optimal control theory. As a consequence, we characterize the singular support of subelliptic wave kernels outside the diagonal. These results show that abnormal extremals play an important role in the classical-quantum correspondence between sub-Riemannian geometry and sub-Laplacians.