Dispersive estimates for Klein-Gordon equations via a physical space approach

TL;DR

This paper extends physical-space dispersive estimates for Klein-Gordon equations using hyperboloidal foliation and frequency truncations, addressing limitations of previous methods and analyzing the vanishing mass limit.

Contribution

It introduces a novel approach combining hyperboloidal foliation with frequency-dependent truncations to obtain dispersive estimates for Klein-Gordon equations.

Findings

Established frequency-restricted $L^1$--$L^ty$ dispersive estimates.

Handled the fixed light-cone limitation of hyperboloidal foliation.

Analyzed the estimates' effectiveness in the vanishing mass limit.

Abstract

Building on the hyperboloidal foliation approach of Lefloch and Ma, we extend Klainerman's physical-space approach to dispersive estimates to recover the frequency-restricted $L^1$--$L^\infty$ dispersive estimates for Klein-Gordon equations. The hyperboloidal foliation approach naturally only provide estimates within a fixed forward light-cone, and is based on an initial data norm that is not translation invariant. Both of these problems can be handled with frequency-dependent physical-space truncations. To handle the lack of scaling symmetry for the Klein-Gordon equation and complete the argument, we also need to keep track of the effectiveness of our estimates in the vanishing mass limit.