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Global parametrices and dispersive estimates for variable coefficient wave equations

Jason Metcalfe, Daniel Tataru

TL;DR

The article develops a global-in-time dispersive framework for variable-coefficient wave equations with time-dependent metrics under a weak asymptotic flatness condition. It combines a paradifferential approach to frequency-localize coefficients, a half-wave decomposition, and a phase-space (FBI/Bargmann) transform framework to construct outgoing parametrices and control errors via localized energy estimates. The main contributions include the existence of frequency-localized parametrices for half-waves, a long-time phase-space parametrix, and the derivation of global Strichartz estimates for small perturbations, together with stable reduction to the half-wave setting. This advances the understanding of long-time dispersion in nontrivial geometries and provides a robust microlocal toolkit for analyzing variable-coefficient wave equations and their dispersive properties.

Abstract

In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity.

Global parametrices and dispersive estimates for variable coefficient wave equations

TL;DR

The article develops a global-in-time dispersive framework for variable-coefficient wave equations with time-dependent metrics under a weak asymptotic flatness condition. It combines a paradifferential approach to frequency-localize coefficients, a half-wave decomposition, and a phase-space (FBI/Bargmann) transform framework to construct outgoing parametrices and control errors via localized energy estimates. The main contributions include the existence of frequency-localized parametrices for half-waves, a long-time phase-space parametrix, and the derivation of global Strichartz estimates for small perturbations, together with stable reduction to the half-wave setting. This advances the understanding of long-time dispersion in nontrivial geometries and provides a robust microlocal toolkit for analyzing variable-coefficient wave equations and their dispersive properties.

Abstract

In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity.

Paper Structure

This paper contains 11 sections, 33 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. A minor simplification
  4. The paradifferential calculus
  5. Localized energy estimates
  6. The half wave decomposition
  7. Parametrices and Strichartz estimates
  8. Pseudodifferential operators and phase space transforms
  9. A long time phase space parametrix
  10. A perturbation of the half wave equation
  11. The parametrix construction

Key Result

Lemma 1

gS a) ($s=0$) We have b) If $0 < s < \frac{n-1}{2}$ then the following Hardy type inequality holds for all $u \in \mathcal{S}({\mathbb R} \times {\mathbb R}^n)$: c) If $\frac{n-1}{2} \leq s < \frac{n+1}{2}$ then we have the weaker bound where the time dependent function $\bar{u}_{A_{<j}}$ stands for the spatial averages of $u$ in $\{ |x| \leq 2^j\}$.

Theorems & Definitions (49)

  • Lemma 1
  • Definition 2
  • Lemma 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Remark 8
  • Lemma 9
  • proof
  • ...and 39 more