Searcharxiv⌕ Search

arXiv subjects

Abdelhamid Boussejra

Publications and source records attributed to Abdelhamid Boussejra.

10 recordsLinked to original sources
arXiv2025

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

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.

math.RT↗
arXiv2024

On Poisson transforms of differential forms on real hyperbolic spaces

This paper is concerned with the Poisson transform of differential forms on the hyperbolic space $H^n(\mathbb R)$. Consider an integer $p$ such that $1\leqslant p\leqslant n$ and let $q$ be either $p-1$ or $p$. For $1<r<\infty$, we prove that the Poisson transform is a topological isomorphism from the space of $L^r$-differential $q$-forms on the boundary $\partial H^n(\mathbb R)$ onto a Hardy-type subspace of $p$-eigenforms of the Hodge-de Rham Laplacian on $H^n(\mathbb R)$.

math.RT↗
arXiv2024

A characterization of the $L^2$-range of the generalized spectral projections related to the Hodge-de Rham Laplacian

Let $H^n(\mathbb R)$ be the real hyperbolic space. In this paper, we present a characterization of the $L^2$-range of the generalized spectral projections on the bundle of differential forms over $H^n(\mathbb R)$. As an underlying result we show a characterization of the $L^2$-range of the Poisson transform on the bundle of differential forms on the boundary $\partial H^n(\mathbb R)$. This gives a positive answer to a conjecture of Strichartz on differential forms.

math.RT↗
arXiv2023

A characterization of the $L^2$-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

We study the $L^2$-boundedness of the Poisson transforms associated to the homogeneous vector bundles $ Sp(n,1)\times_{Sp(n)\times Sp(1)} V_τ$ over the quaternionic hyperbolic spaces $ Sp(n,1)/Sp(n)\times Sp(1)$ associated with irreducible representations $τ$ of $ Sp(n)\times Sp(1)$ which are trivial on $ Sp(n)$. As a consequence, we describe the image of the section space $L^2(Sp(n,1)\times_{Sp(n)\times Sp(1)} V_τ)$ under the generalized spectral projections associated to a family of eigensections of the Casimir operator.

math.RT↗
arXiv2022

On Poisson transform for spinors

Let $(τ,V_τ)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(σ,V_σ)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $τ_{\mid \mathrm{Spin}(n-1)}$. We consider the real hyperbolic space $H^n(\mathbb R)$ as the rank one homogeneous space $\mathrm{Spin}_0(1,n)/\mathrm{Spin}(n)$ and the spinor bundle $ΣH^n(\mathbb R)$ over $H^n(\mathbb R)$ as the homogeneous bundle $\mathrm{Spin}_0(1,n)\times_{\mathrm{Spin}(n)} V_τ$. Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on $ΣH^n(\mathbb R)$ which can be written as the Poisson transform of $L^p$-sections of the bundle $\mathrm{Spin}(n)\times_{\mathrm{Spin}(n-1)} V_σ$ over the boundary $S^{n-1}\simeq \mathrm{Spin}(n)/\mathrm{Spin}(n-1)$ of $H^n(\mathbb R)$, for $1<p<\infty$.

math.RT↗
arXiv2021

The Strichartz conjecture for the Poisson transform on homogeneous line bundles over Noncompact Complex Grassmann manifolds

Let \(X=G/K\) be a noncompact complex Grassmann manifold of rank \(r\). Let \(τ_l\) be a character of \(K\), \(G\times_P{\C}\) and \(G\times_K{\C}\) the homogeneous line bundles associated with the representations \(σ_{λ,l}=τ_l\otimes a^{ρ-iλ}\otimes 1\) of \(P=MAN\) and \(τ_l\) of \(K\). We give an image characterization for the Poisson transform \(P_{λ,l}\) of \\\(L^2\)-sections of the unitary principal series representations of \(G\) parametrized by \(σ_{λ,l}\). More precisely for real and regular parameter \(λ\) in \(\mathfrak{a}^\ast\) we prove that \(P_{λ,l}\) is an isomorphism from \(L^2(K\times_M{\C})\) onto the space of joint eigensections \(F\) of the algebra of \(G\)-invariant differential operators on \(G\times_K{\C}\) that satisfy the following growth condition \begin{eqnarray*} \sup_{R>1}\frac{1}{R^r}\int_{B(R)}\mid F(g)\mid^2\, {\rm d}g<\infty. \end{eqnarray*} This generalizes a conjecture by Strichartz which corresponds to \(τ_l\) trivial.

math.RT↗
arXiv2019

L^p-Poisson integral representations of the generalized Hua operators on line bundles over SU(n,n)/S(U(n)xU(n))

Let $τ_ν$ ($ν\in \mathbb{Z}$) be a character of $K=S(U(n)\times U(n))$, and $SU(n,n)\times_K\mathbb{C}$ the associated homogeneous line bundle over $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. Let $\mathcal{H}_ν$ be the Hua operator on the sections of $SU(n,n)\times_K\mathbb{C}$. Identifying sections of $SU(n,n)\times_K\mathbb{C}$ with functions on $\mathcal{D}$ we transfer the operator $\mathcal{H}_ν$ to an equivalent matrix-valued operator $\widetilde{\mathcal{H}}_ν$ which acts on $\mathcal{D}$ . Then for a given ${\mathbb{C}}$-valued function $F$ on $\mathcal{D}$ satisfying $\widetilde{\mathcal{H}}_νF=-\frac{1}{4}(λ^2+(n-ν)^2) F.(\begin{smallmatrix} I&0 0&-I \end{smallmatrix})$ we prove that $F$ is the Poisson transform by $P_{λ,ν}$ of some $f\in L^p(S)$, when $1 n-1$. This generalizes the result in \cite{B1} which corresponds to $τ_ν$ the trivial representation.

math.RT↗