Boundary value problems for the Shilov boundary of a bounded symmetric domain of tube type
Let \(X=G/K\) be a Riemannian symmetric space of noncompact type. \textit{M. Kashiwara}, \textit{A. Kowata}, \textit{K. Minemura}, \textit{K. Okamoto}, \textit{T. Oshima} and \textit{M. Tanaka} [Ann. Math. (2) 107, 1--39 (1978; Zbl 0377.43012)] proved Helgason's conjecture: The Poisson transform \(P_\lambda\) is a \(G\)-isomorphism of \(B(G/P, L_\lambda)\) onto \(A(G/K, M_\lambda)\) under the condition that \(-2\langle \lambda,a\rangle/ \langle\alpha, \alpha\rangle \notin \{1,2, \dots\}\). Here \(B(G/P,L_\lambda)\) is the space of all hyperfunction valued sections of a principal series attached to the Furstenberg boundary \(G/P\) of \(X\) and \(A(G/K,M_\lambda)\) is the space of all real analytic functions on \(G/K\) satisfying the system of differential equations \(M_\lambda\). The author assumes that \(X\) is a Hermitian symmetric space of tube type of rank greater than one, and considers a characterization of \(P_\lambda (B(G/P_\Xi,s))\), where \(B(G/P_\Xi,s)\) \((s\in\mathbb{C})\) is a \(G\)-submodule of \(B(G/P,L_{\lambda_s})\) which consists of all hyperfunction valued sections of a degenerate series attached to the Shilov boundary \(G/P_\Xi\) of \(X\). He shows that, if \(\lambda_s\) satisfies the same condition as above, the Poisson transform \(P_{\lambda_s}\) is a \(G\)-isomorphism of \(B(G/P_\Xi,s)\) onto \(A(G/K,N_s)\), where \(N_s\) is the system of the Hua equations. When \(\lambda_s=\rho\), there are several works in this direction by various authors. The essential part of the proof is the surjectivity: For \(u\in A(G/K,N_s)\) the boundary value \(\beta_{\lambda_s} (u)\) belongs to \(B(G/P_\Xi,s)\). To deduce this, by applying \textit{T. Oshima}'s theory of the boundary map [Adv. Stud. Pure Math. 4, 391--432 (1984; Zbl 0576.58030)] and by using a similar calculation done by \textit{M. Lassalle} [Invent. Math. 77, 129--161 (1984; Zbl 0582.32042)], the author obtains the induced equation that \(\beta_{\lambda_s} (u)\) satisfies.
scientific article; zbMATH DE number 933300
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