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Strichartz's conjecture for the spinor bundle over the real hyperbolic space

Abdelhamid Boussejra, Khalid Koufany

TL;DR

The paper advances harmonic analysis on the spinor bundle over real hyperbolic space by extending Strichartz's conjecture to the spinor setting. It develops a full Helgason–Fourier framework for $L^2$ spinor sections, establishes uniform $L^2$ bounds for Poisson transforms and spectral projections, and proves that these transforms provide isomorphisms onto natural joint-eigenfunction spaces. A key contribution is the derivation of precise asymptotic expansions (scattering formulas) for $\tau$-spherical functions and their use in proving both Poisson-transform and spectral-projection conjectures. The results deepen the connection between Poisson transforms, Eisenstein integrals, and Dirac-type operators on $H^n(\,\mathbb{R})$, with potential implications for spectral theory and vector-valued harmonic analysis on symmetric spaces.

Abstract

Let $H^n(\mathbb R)$ denote the real hyperbolic space realized as the symmetric space $Spin_0(1,n)/Spin(n)$. In this paper, we provide a characterization for the image of the Poisson transform for $L^2$-sections of the spinor bundle over the boundary ${\partial H}^n(\mathbb R)$. As a consequence, we obtain an $L^2$ uniform estimate for the generalized spectral projections associated to the spinor bundle over $H^n(\mathbb R)$, thereby extending Strichartz's conjecture from the scalar case to the spinor setting.

Strichartz's conjecture for the spinor bundle over the real hyperbolic space

TL;DR

The paper advances harmonic analysis on the spinor bundle over real hyperbolic space by extending Strichartz's conjecture to the spinor setting. It develops a full Helgason–Fourier framework for spinor sections, establishes uniform bounds for Poisson transforms and spectral projections, and proves that these transforms provide isomorphisms onto natural joint-eigenfunction spaces. A key contribution is the derivation of precise asymptotic expansions (scattering formulas) for -spherical functions and their use in proving both Poisson-transform and spectral-projection conjectures. The results deepen the connection between Poisson transforms, Eisenstein integrals, and Dirac-type operators on , with potential implications for spectral theory and vector-valued harmonic analysis on symmetric spaces.

Abstract

Let denote the real hyperbolic space realized as the symmetric space . In this paper, we provide a characterization for the image of the Poisson transform for -sections of the spinor bundle over the boundary . As a consequence, we obtain an uniform estimate for the generalized spectral projections associated to the spinor bundle over , thereby extending Strichartz's conjecture from the scalar case to the spinor setting.
Paper Structure (9 sections, 12 theorems, 157 equations)

This paper contains 9 sections, 12 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The Helgason Fourier restriction theorem
  4. Poisson transform on spinors
  5. Announcement of the main results
  6. Asymptotic expansion for $\tau$-spherical functions
  7. Proof of Theorem \ref{['main-th-Poisson']} -- Strichartz's conjecture for Poisson transform
  8. Proof of Theorem \ref{['main-th-proj']} - Strichartz's conjecture for spectral projections
  9. Appendix

Key Result

Theorem 3.1

Let $\tau\in\Lambda$ and $\sigma\in\widehat{M}(\tau)$. $(1)$ Suppose $n$ even. There exists a positive constant $C$ such that for $\lambda\in\mathbb R\setminus\{0\}$ and $R>1$ we have for every $f\in L^2(G,\tau)$ supported in $B(R)$. $(2)$ Suppose $n$ odd. There exists a positive constant $C$ such that for $\lambda\in\mathbb R\setminus\{0\}$ and $R>0$ we have for every $f\in L^2(G,\tau)$ support

Theorems & Definitions (18)

  • Theorem 3.1: Helgason Fourier restriction theorem
  • proof
  • Proposition 4.1
  • Theorem 5.1: Strichartz's conjecture for the spectral projections
  • Theorem 5.2: Strichartz's conjecture for the Poisson transform
  • Proposition 6.1
  • proof
  • Proposition 6.2: Key formula
  • proof
  • Lemma 6.3
  • ...and 8 more