Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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Journal of Functional Analysis 236 (2006) 546–580 www.elsevier.com/locate/jfa Hua operators and Poisson transform for bounded symmetric domains ✩ Khalid Koufany a , Genkai Zhang b,∗ a Institut Élie Cartan, UMR 7502, Université Henri Poincaré (Nancy 1), BP 239, F-54506 Vandœuvre-lès-Nancy cedex, France b Department of Mathematics, Chalmers University of Technology and Göteborg University, S-41296 Göteborg, Sweden Received 8 December 2005; accepted 24 February 2006 Available online 4 April 2006 Communicated by Paul Malliavin Abstract Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z)for a hyperfunction f on S defined by the Poisson kernel Ps(z, u) = (h(z, z) n/r /|h(z, u) n/r | 2 ) s , (z, u)×Ω ×S, s ∈ C.Forallssatisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image. © 2006 Elsevier Inc. All rights reserved. Keywords: Bounded symmetric domains; Shilov boundary; Invariant differential operators; Eigenfunctions; Poisson transform; Hua systems 1. Introduction Let Ω = G/K be a Riemannian symmetric space. Any parabolic subgroup P of G defines a boundary G/P of the symmetric space Ω. The Poisson transform is an integral operator from hyperfunctions on G/P into the space of eigenfunctions on Ω of the algebra D(Ω) G of invariant ✩ Research of the authors is partly supported by European IHP network Harmonic Analysis and Related Problems. Research of G. Zhang is supported by Swedish Science Council (VR). * Corresponding author. E-mail addresses: khalid.koufany@iecn.u-nancy.fr (K. Koufany), genkai@math.chalmers.se (G. Zhang). 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.02.014
K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 547 differential operators. Any such boundary G/P can be viewed as coset space of the maximal boundary G/Pmin defined by a minimal parabolic subgroup Pmin. In this case the most general result was obtained by Kashiwara et al. [7] where they proved that under certain conditions on the eigenvalues the Poisson transform is a G-isomorphism between the space of hyperfunctions on G/Pmin and the space of eigenfunctions of invariant differential operators on Ω, namely the Helgason conjecture. It thus arises the question of characterizing the image of the Poisson transform for other smaller boundaries. Suppose Ω is a bounded symmetric domain in a complex n-dimensional space V .LetSbe its Shilov boundary and r its rank. In this paper we consider the characterization of the image of the Poisson transform Psϕ(z) = Ps(z, u)ϕ(u) dσ (u) on the Shilov boundary S when s satisfies the following condition: −4 b + 1 + j a 2 S n + (s − 1) /∈{1, 2, 3,...} for j = 0 and 1, (1) r where a and b are some structure constants of Ω. For a specific value of s (s = 1 in our parameterization) the kernel Ps(z, u), (z, u) ∈ Ω × S, is the so-called Poisson kernel for harmonic functions, and the corresponding Poisson transform P := P1 maps hyperfunctions on S to harmonic functions on Ω; here harmonic functions are defined as the smooth functions that are annihilated by all invariant differential operators that annihilate the constant functions. When Ω is a tube domain Johnson and Korányi [6] proved that the image of the Poisson transform P is exactly the set of all Hua-harmonic functions. For non-tube domains the characterization of the image of the Poisson transform P was done by Berline and Vergne [1] where certain third-order differential Hua operator was introduced to characterize the image. In his paper [12] Shimeno considered the Poisson transform Ps on tube domains; it is proved that Poisson transform maps hyperfunctions on the Shilov boundary to certain solution space of the Hua operator. For general domains and for other boundaries, the image of the Poisson transform was characterized in [13]. However for the Shilov boundary of a non-tube domain the problem is still open. We will construct two Hua operators of the third order and use them to give a characterization. For the matrix ball Ir,r+b of r × (r + b)-matrices some eigenvalue equation for the second-order Hua operator (constructed by Hua [5] and reformulated by Berline and Vergne [1]) is proved to give the characterization. We proceed to explain the content of our paper. Hua operator of the second-order H for a general symmetric domain is defined as a kC-valued operator, see Section 4. For tube domains it maps the Poisson kernels into the center of kC, namely the Poisson kernels are its eigenfunctions up to an element in the center, but it is not true for non-tube domains, see Section 5. However for type I domains of non-tube type, see Section 6, there is a variant of the Hua operator, H (1) , by taking the first component of the operator, since in this case kC = k (1) C + k(2) C is a sum of two irreducible ideals. We prove that operator H (1) has the Poisson kernels as its eigenfunctions and we find the eigenvalues. We prove further that the eigenfunctions of the Hua operator H (1) are also eigenfunctions of invariant differential operators on Ω. For that purpose we compute the radial part of Hua operator H (1) ,
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When G = η ⊗ r, we obtain Becaus
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we now have (cf. (7.8)) that H. Sch
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