Massless Dirac equation on spinor bundles over real hyperbolic spaces

TL;DR

This paper establishes sharp dispersive and Strichartz estimates for the Dirac equation on spinor bundles over real hyperbolic spaces, revealing the influence of negative curvature on wave dispersion and extending known results.

Contribution

It provides the first sharp-in-time dispersive estimate for the Dirac equation on hyperbolic spaces, highlighting differences from Euclidean cases and improving global Strichartz estimates.

Findings

Dispersive estimates differ between short and long times due to negative curvature.

The Euclidean equivalence between Dirac and wave propagators does not hold in hyperbolic space.

Global Strichartz estimates are improved with no loss of angular derivatives.

Abstract

We prove a sharp-in-time dispersive estimate of the Dirac equation on spinor bundles over the real hyperbolic space. Compared with the Euclidean counterparts, our result shows that the dispersive estimate differs between short and long times, reflecting the intuitive influence of negative curvature on the dispersion. Moreover, the well-known equivalence between dispersive estimates for Dirac and wave propagators in the Euclidean setting no longer holds in this context. This finding suggests that spinor fields are affected by the geometry at infinity of the manifold. As a key application, we establish an improved global-in-time Strichartz estimate, in the sense that there is no loss of angular derivatives and the admissible set is larger than previously known results in other settings.