\(L^{2}\)-Poisson integral representations of solutions of the Hua system on the bounded symmetric domain \(SU(n,n)\)/\(S(U(n){\times}U(n))\)
Let \(D= \{Z\in M(n, \mathbb{C})\mid I_n- ZZ^*\) positive definite\} be the matrix ball of rank \(n\) which is a model for the bounded symmetric domain \(SU(n,n)/S(U(n)\times U(n))\). The Shilov boundary \(S\) of \(D\) is the unitary group \(U(n)\) of \(\mathbb{C}^n\). Let \(\partial_Z\) denote the \(n\times n\) matrix of differential operators \(({\partial\over\partial z_{i,j}})_{i,j}\) if \(Z= (z_{i,j})_{i,j}\in D\). The Hua operator associated to the matrix ball \(D\) is \[ H_D= 2[(I_n- ZZ^*)\overline\partial_Z(I_n- Z^*Z) \partial_Z'+ \partial_Z'(I_n- ZZ^*)\overline\partial_Z(I_n- Z^* Z)], \] where \(\overline\partial_Z\) and \(\partial_Z'\) denote the conjugate and the transpose of the matrix operator \(\partial_Z\), respectively (with the understanding that \(\overline\partial_Z\) and \(\partial_Z'\) do not differentiate the matrices \(I_n- Z^*Z\) and \(I_n- ZZ^*\)). For \(\lambda\in\mathbb{C}\) with \(\text{Re}(i\lambda)> n-1\) the class \(E_\lambda(D)\) of solutions \(F\) of the following Hua system of differential equations on \(D\) is considered: \(H_DF(Z)= -(\lambda^2+ n^2) F(Z)\cdot I_n\). It is shown that such an \(F\) is the Poisson integral \[ P_\lambda f(Z)= \int_S \Biggl[{\text{det}(I_n- ZZ^*)\over |\text{det}(I_n- ZU^*)|^2}\Biggr]^{{i\lambda+ n\over 2}} f(U) dU \] of an \(L^2\)-function \(f\) on \(S\) if and only if it satisfies the following growth condition of Hardy type: \[ \sup_{0\leq r< 1} \Biggl[(1- r^2)^{-{n(n- \text{Re}(i\lambda))\over 2}}\Biggl(\int_S|F(rU)|^2 dU\Biggr)^{{1\over 2}}\Biggr]< +\infty. \] The proof relies on the asymptotic behaviour of the generalized spherical functions \(\Phi_{\lambda, m}(r)\), which are eigenvalues of the operator \[ f\mapsto (U\mapsto P_\lambda f(rU))\qquad (f\in L^2(S),\;r\in [0,1[). \] However, in the proof of the technical Lemma 4.1 the last assertion on the asymptotic behaviour of the determinants of hypergeometric functions is not sufficiently explained.
scientific article; zbMATH DE number 1974789
scientific article