Abdelhamid Boussejra

Université Ibn Tofail · Department of Mathematics

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_\tau$ over the quaternionic hyperbolic spaces $Sp(n,1)/Sp(n)\times Sp(1)$ associated with irreducible representations $\tau$ of $Sp(n)\times Sp(1)$ which are trivial on $Sp(n)$. As a consequence, we describe t...

Let $(\tau,V_\tau)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(\sigma,V_\sigma)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $\tau_{\mid \mathrm{Spin}(n-1)}$. 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...

Let \(E_l=G\times _K{{\mathbb {C}}}\) be the associated homogeneous line bundle to a one-dimensional \(K\)-representation \(\tau _l\) (\(l\in {{\mathbb {Z}}}\)) over the noncompact complex Grassmann manifold \(G/K\); \(G=SU(r,r+b)\) and \(K=S(U(r)\times U(r+b))\). Let \({\mathbb {D}}(E_l)\) be the algebra of \(G\)-invariant differential operators o...

We study the Poisson transform for differential forms on the real hyperbolic space $\mathbb H^n$. For $1<r<\infty$, we prove that the Poisson transform is an isomorphism from the space of $L^r$ differential $q$-forms on the boundary $\partial \mathbb H^n$ onto a Hardy-type subspace of $p$-eigenforms of the Hodge-de Rahm Laplacian, for $0\leq p<\fra...

Let τν (ν∈Z) be a character of K=S(U(n)×U(n)), and SU(n,n)×KC the associated homogeneous line bundle over D={Z∈M(n,C):I−ZZ⁎>0}. Let Hν be the Hua operator on the sections of SU(n,n)×KC. Identifying sections of SU(n,n)×KC with functions on D we transfer the operator Hν to an equivalent matrix-valued operator H˜ν which acts on D. Then for a given C-v...

Let \(X=G/K\) be a noncompact complex Grassmann manifold of rank \(r\). Let \(\tau_l\) be a character of \(K\), \(G\times_P{\C}\) and \(G\times_K{\C}\) the homogeneous line bundles associated with the representations \(\sigma_{\lambda,l}=\tau_l\otimes a^{\rho-i\lambda}\otimes 1\) of \(P=MAN\) and \(\tau_l\) of \(K\). We give an image characterizati...

Let $\tau_\nu$ ($\nu \in \mathbb{Z}$) 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 $\mathcal{H}_\nu$ be the Hua operator on the sections of $SU(n,n)\times_K\mathbb{C}$. Identifying sections of $SU(n,n)\times_K\mathbb{C}$...

Let $ B(\mathbb{O}^2)=\{x\in \mathbb{O}^2,|x|<1\}$ be the bounded realization of the exceptional symmetric space $F_{4(-20)}/Spin(9)$. For a non-zero real number $\lambda$, we give a necessary and a sufficient condition on eigenfunctions $F$ of the Laplace-Beltrami operator on $B(\mathbb{O}^2)$ with eigenvalue $-(\lambda^2+\rho^2)$ to have an $L^p$...

We prove that every representation of the Cuntz algebra on a separable Hilbert space H arises from a pure isometry V whose wandering space has dimension N. We identify the permutative representations in this construction. See https://doi.org/10.1016/j.jmaa.2017.04.031

The Laplacian and Ornstein–Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U(n,1)U(n,1). We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to...

Let D be a bounded symmetric domain of tube type . We show that the image of the Poisson transform on the degenerate principal series representation attached to the Shilov boundary of D is characterized by a covariant differential operator on a homogeneous line bundle on D.

In this paper we investigate the boundary behavior of Lp -Poisson integrals for various boundaries of Riemannian Symmetric Spaces of the non-compact type. In particular, we show that if a function F on a Riemannian symmetric space G=K is solution of some invariant differential system associated to a standard parabolic subgroup PE of G then F is the...

We characterize the $L^p$-range, $1 < p < +\infty$, of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains. We prove that this range is a Hua-Hardy type space for harmonic functions satisfying a Hua system.

In this paper, we give a necessary and sufficient condition on eigenfunctions of the Hua operator on a Hermitian symmetric space of tube type X=G/K, to have an Lp-Poisson integral representations over the Shilov boundary of X. More precisely, let λ∈C such that R(λ)>η−1 (2η being the genus of X) and let F be a C-valued function on X satisfying the f...

Let positive definite} be the matrix ball of rank n and let HD be the associated Hua operator. For a complex number λ, such that Reiλ>n−1 we give a necessary and sufficient condition on solutions F of the following Hua system of differential equations on D:to have an L2-Poisson integral representations over the Shilov boundary S of D. Namely, F is...

The aim of this paper is to give, in a unified manner, the characterization of the L-P-range (p greater than or equal to 2) of the Poisson transform P-lambda for the Hyperbolic spaces B(F-n) over F = R, C or the quaternions H. Namely, if Delta is the Laplace-Beltrami operator of B(F-n) and sF a C-valued function on B(F-n) satisfying DeltaF = (lambd...

For every fixed real numberλrelated to the continuous spectrum of the invariant LaplaciansΔαβ=4(1−|z|2)∑i,j=1n(δij−zizj)∂2∂zj∂zj+α∑j=1nzj∂∂zj+β∑j=1nzj∂∂zj−αβinBn, we characterize the eigenfunctions ofΔαβthat are Poisson integrals ofL2-functions on the boundary ofBn. These eigenfunctions occurred in the weighted Plancherel formula of the unit comple...

We give a characterization of those solutions F of the Hua system of differential equations HDF(Z) = − (λ2 + 4)F(Z) · I, that are Poisson-Shilov integrals of L2-functions on the Shilov boundary of the rank two matrix ball.

We give a characterization of those solutions F of the Hua system of differential equations HDF(Z) = -(X2 + 4)F(Z) · I, that are Poisson-Shilov integrals of L2-functions on the Shilov boundary of the rank two matrix ball.