The Hua operators on homogeneous line bundle on Bounded Symmetric Domains of Tube Type

Abstract

Let D = G/K be a bounded symmetric domain of tube type. We show that the image of the Poisson transform on the degenerate principal series representation of G attached to the Shilov boundary of D is characterized by a K-covariant differential operator on a homogeneous line bundle over D. As a consequence of our result we get the eigenvalues of the Casimir operator for Poisson transforms on homogeneous line bundles over G/K. This extends a result of Imemura and all [5] on symmetric domains of classical type to all symmetric domains. Also we compute a class of Hua type integrals generalizing an earlier result of Faraut and Koranyi [3].

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

“…As a direct consequence of Proposition 3.1 and the main theorem in [4] we get explicitly the series expansion of the eigenfunctions of the generalized Hua operator.…”

“…The operator H ν will be called here the generalized Hua operator. In [8] Koufany and Zhang proved that the Poisson transform P λ,ν is an isomorphism from B(G/P Ξ , σ λ,ν ) onto an eigenspace E λ,ν (G/K, τ ν ) of the generalized Hua operator, under certain condition on λ (see also [[10], [4]]). This result suggests the problem of characterizing the Poisson transforms P λ,ν (B 0 (S)) where B 0 (S) is some natural subspace of B(G/P Ξ , σ λ,ν ) such as the space L p (G/P Ξ , σ λ,ν ) ....…”

“…Since tr((I − ZZ * )∂(I − Z * Z)∂ ′ − ν(I − ZZ * )∂Z * ) is -up to a constant-the Casimir operator of SU(n, n) with respect to a weighted action with weight ν ∈ Z, it follows that the elements of E λ,ν (D) are real analytic functions on D, see [6]. Moreover, for ν ∈ Z and λ ∈ C such that iλ / ∈ 2Z − + n − 2 ± ν, the solutions F of (2) are given by generalized Poisson transforms of hyperfunctions f on the Shilov boundary, see [8] and [4]. In this paper we will show that the solutions of the generalized Hua system (2) for which the hyperfunction f is given by a Borel measure or an L p -function (1 < p < ∞) are characterized by an H p -condition.…”