Massive wave propagation near null infinity

Abstract

We study, fully microlocally, the propagation of massive waves on the octagonal compactification \[\mathbb{O}=[\overline{\mathbb{R}^{1,d}};\mathscr{I};1/2]\] of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti-Shubin-Melrose's sc-calculus) and, more novelly, at null infinity, denoted $\mathscr{I}$. The analysis is closely related to Hintz-Vasy's recent analysis of massless wave propagation at null infinity using the ``e,b-calculus'' on $\mathbb{O}$. We prove several elementary corollaries regarding the Klein-Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, $\Psi_{\mathrm{de,sc}}(\mathbb{O})$, the ``de,sc-calculus'' on $\mathbb{O}$. The `de' refers to the structure (``double edge'') of the calculus at null infinity, and the `sc' refers to the structure (``scattering'') at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein-Gordon equation. Unlike hyperbolic coordinates, the de,sc-boundary fibration structure is Poincar\'e invariant.