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Wave equation on general noncompact symmetric spaces

Jean-Philippe Anker, Hong-Wei Zhang

Abstract

We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Wave equation on general noncompact symmetric spaces

Abstract

We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Paper Structure

This paper contains 19 sections, 27 theorems, 291 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Spherical Fourier analysis on $\mathbb{X}$
  5. Barycentric decomposition of the Weyl chamber
  6. Pointwise estimates of the wave kernel
  7. Estimates of $\widetilde{\omega}_{t}^{\sigma,0}(x)$ when $|t|$ is large and $\frac{|x|}{|t|}$ is sufficiently small.
  8. Estimates of $\widetilde{\omega}_{t}^{\sigma,0}(x)$ in the remaining range
  9. Estimates of $\omega_{t}^{\sigma,\infty}$
  10. Dispersive estimates
  11. Strichartz inequality and applications
  12. Strichartz inequality
  13. Global well-posedness in $L^{p}(\mathbb{R},L^q(\mathbb{X}))$
  14. Further results for Klein-Gordon equations
  15. Oscillatory integral on $\mathfrak{a}$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $d\ge3$ and $\sigma\in\mathbb{C}$ with $\mathop{\mathrm{Re}}\nolimits\sigma=\frac{d+1}{2}$. There exist $C>0$ and $N\in\mathbb{N}$ such that the following estimates hold for all $t\in\mathbb{R}^{*}$ and $x\in\mathbb{X}$: where $x^{+}\in\overline{\mathfrak{a}^{+}}$ denotes the radial component of $x$ in the Cartan decomposition, and $D=\ell+2|\Sigma_{r}^+|$ is the so-called dimension at infini

Figures (4)

  • Figure 1: Examples of barycentric subdivisions in $A_2$ and in $A_3$.
  • Figure 2: Example of the projection in $A_3$
  • Figure 3: Admissibility in dimension $d \ge 4$.
  • Figure 4: Extended admissibility in dimension $d\ge4$.

Theorems & Definitions (62)

  • Theorem 1.1: Pointwise kernel estimates
  • Remark 1.2
  • Theorem 1.3: Dispersive property
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 52 more