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Function spaces and reproducing kernels on bounded symmetric domains. (English) Zbl 0718.32026
Let D be an irreducible bounded symmetric domain in the Harish-Chandra realization, i.e., \(D=G\cdot O\simeq /K\subset p^+\). There are several natural Hilbert spaces of holomorphic functions on D, such as the Bergman-type spaces and also the Hardy type spaces. The space of holomorphic polynomials on \(p^+\) decomposes into irreducible subspaces under the action of the isotropy group K of D, i.e., \[ {\mathcal P}(p^+)=\oplus_{m\geq 0}{\mathcal P}_ m(p^+). \] Each irreducible subspace contains a unique normalized L-invariant “spherical” polynomial \(\phi_ m\), where L is the isotropy group of the Shilov boundary in K.
One of the main results of the paper is the explicit computation of the norms of the spherical polynomials \(\{\phi_ m\}\) in each of the Hilbert spaces considered. The authors also obtain a description of the reproducing kernels of the K-irreducible subspaces in each of the Hilbert spaces, and an expansion in terms of these form all complex powers of the Bergman kernel.
The results have several applications, including the following: For \(\lambda >p-1\), (p a constant depending on the domain D) let \(L^ 2_{\lambda}\) denote the Hilbert space of holomorphic functions f on D for which \[ \| f\|^ 2_{\lambda}= c_{\lambda}\int_{D}| f(z)|^ 2h(z)^{\lambda -p}dz<\infty. \] Here h(z) is the function on D given by \(h(z)^{-p}=| J_ g(0)|^ 2\), where \(J_ g\) is the complex Jacobian determinant of an element \(g\in G\) such that \(g\cdot z=0\). Let \(K_{\lambda}\) be the reproducing kernel of \(L^ 2_{\lambda}\) and let h(z,w) be the function satisfying \(h(z,w)^{- \lambda}=K_{\lambda}(z,w)\). For \(\gamma\in {\mathbb{R}}\), \(\lambda >p-1,\) and \(z\in D\), define \[ I_{\gamma}(z)=\int_{S}| h(z,u)|^{- (n/r+\gamma)}du,\quad J_{\gamma,\lambda}(z)=\int_{D}| h(z,w)|^{-(\lambda +\gamma)}h(w,w)^{\lambda -p}dw. \] The authors prove that if \(\gamma >(r-1)a/2\), then \[ I_{\gamma}(z)\approx J_{\gamma,\lambda}(z)\approx h(z,z)^{-\gamma}. \] This is a generalization of some inequalities on the unit ball in \({\mathbb{C}}^ n\) due to F. Forelli and W. Rudin [Indiana Univ. Math. J., 24, 593-602 (1974; Zbl 0297.47041)]. This result can also be used to generalize the results of Forelli and Rudin concerning projection operators onto \(L^ p\)-spaces of holomorphic functions.
Reviewer: M.Stoll (Columbia)

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
Full Text: DOI
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