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Shifted wave equation on noncompact symmetric spaces

Yulia Kuznetsova, Zhipeng Song

TL;DR

The article analyzes the shifted wave equation on noncompact symmetric spaces by inspecting oscillatory functions of the shifted Laplacian $\Delta_\rho$ through spherical Fourier analysis. It develops a refined stationary-phase framework for oscillatory integrals with a radial factor to obtain precise pointwise kernel bounds, including the delicate case with no spectral gap. The main results provide explicit time-decay and spatial-weighted bounds for the kernel in several regimes determined by a smoothing parameter, and deduce dispersive $L^{p'}-L^p$ estimates via Kunze-Stein, extending prior rank-one results to higher-rank symmetric spaces. These bounds establish a foundation for further Strichartz-type estimates for wave-type equations on $G/K$ and have potential applications in harmonic analysis and PDE on noncompact symmetric spaces.

Abstract

Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We consider the Laplace-Beltrami operator $Δ$ on the homogeneous space $G/K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp( it\sqrt{|x|}) ψ(x) $ applied to the shifted Laplacian $Δ+|ρ|^2$, a case not available before. We obtain a polynomial decay in time of the kernel, and of the $L^{p'}-L^p$ norms of the operator, for $2<p<\infty$.

Shifted wave equation on noncompact symmetric spaces

TL;DR

The article analyzes the shifted wave equation on noncompact symmetric spaces by inspecting oscillatory functions of the shifted Laplacian through spherical Fourier analysis. It develops a refined stationary-phase framework for oscillatory integrals with a radial factor to obtain precise pointwise kernel bounds, including the delicate case with no spectral gap. The main results provide explicit time-decay and spatial-weighted bounds for the kernel in several regimes determined by a smoothing parameter, and deduce dispersive estimates via Kunze-Stein, extending prior rank-one results to higher-rank symmetric spaces. These bounds establish a foundation for further Strichartz-type estimates for wave-type equations on and have potential applications in harmonic analysis and PDE on noncompact symmetric spaces.

Abstract

Let be a semisimple, connected, and noncompact Lie group with a finite center. We consider the Laplace-Beltrami operator on the homogeneous space by a maximal compact subgroup . We obtain pointwise estimates for the kernel of an oscillating function applied to the shifted Laplacian , a case not available before. We obtain a polynomial decay in time of the kernel, and of the norms of the operator, for .
Paper Structure (14 sections, 23 theorems, 218 equations)

This paper contains 14 sections, 23 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Symmetric spaces
  4. Spherical Fourier analysis
  5. Distinguished Laplacian
  6. The method of stationary phase
  7. Oscillating integrals with a radial factor
  8. Stationary phase analysis for a fixed radius
  9. Estimates of derivatives in $u$
  10. Estimates of derivatives in $r$
  11. Global estimates for oscillatory functions of the shifted Laplacian
  12. Some Lemmas
  13. Proof of Theorem \ref{['eit-psi-LB-no-gap']}
  14. $L^p\to L^{p'}$ estimates

Key Result

Theorem 1

AnkerZhang Set $\psi(x) = |x|^{-\alpha}$ with $\Re \alpha \ge(n+1)/2$; then the kernel $k_t$ of $\Psi(\Delta)$ satisfies

Theorems & Definitions (41)

  • Theorem
  • Theorem : \ref{['eit-psi-LB-no-gap']}
  • Proposition 2.1
  • proof
  • Theorem 2.2: gangolli1988harmonic
  • Proposition 2.3
  • proof
  • Theorem 2.5: gangolli1988harmonic
  • Theorem 2.6
  • Remark 3.2
  • ...and 31 more