Massive Wave Propagation Near Null Infinity

Abstract We study, fully microlocally, the propagation of massive waves on the octagonal compactification $$\begin{aligned} \mathbb {O}=[\overline{\mathbb {R}^{1,d}};\mathscr {I};1/2] \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>=</mml:mo> <mml:mo>[</mml:mo> <mml:mover> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>;</mml:mo> <mml:mi>I</mml:mi> <mml:mo>;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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 $$\mathscr {I}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> . The analysis is closely related to Hintz–Vasy’s recent analysis of massless wave propagation at null infinity using the “e,b-calculus” on $$\mathbb {O}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> . We prove several elementary corollaries regarding the Klein–Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, $$\Psi _{\textrm{de,sc}}(\mathbb {O})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ψ</mml:mi> <mml:mtext>de,sc</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>O</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , the “de,sc-calculus” on $$\mathbb {O}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> . 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 Poincaré invariant.