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Local smoothing and Hardy spaces for Fourier integral operators on manifolds

Naijia Liu, Jan Rozendaal, Liang Song, Lixin Yan

Abstract

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all $2<p<\infty$ and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces.

Local smoothing and Hardy spaces for Fourier integral operators on manifolds

Abstract

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of -based Sobolev spaces.
Paper Structure (42 sections, 43 theorems, 240 equations)

This paper contains 42 sections, 43 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Local smoothing for wave equations
  4. Nonlinear wave equations
  5. Local smoothing for Fourier integral operators
  6. Wolff-type inequalities
  7. Hardy spaces for Fourier integral operators
  8. Organization
  9. Notation and terminology
  10. Hardy spaces for Fourier integral operators on $\mathbb{R}^{n}$
  11. Definitions
  12. Basic properties
  13. Boundedness of Fourier integral operators
  14. Sequence spaces
  15. Definitions
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $p\in(2,\infty)$ and $\varepsilon,t_{0}>0$. Then there exists a $C\geq0$ such that, for all $u_{0}\in \mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(M)$ and $u_{1}\in \mathcal{H}^{d(p)-s(p)-1+\varepsilon,p}_{FIO}(M)$, the solution $u$ to eq:Cauchyintro satisfies

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 90 more