Estimates of an integral operator on function spaces (Q1417167)

Let \(D_n = \{(z_1, \dots, z_n) \in \mathbb C^n: | z_j| < 1\), \(j=1, \dots, n\}\). Assume that \(g_j\), \(j=1, \dots, n\), are analytic functions on the unit disk in the complex plane \(\mathbb C\). Define the operator \(T_{\vec g}\) by \[ T_{\vec g} (f) (z) = \int^{z_1}_0 \dots \int^{z_n}_0 f (\zeta_1, \dots,\zeta_n) \prod^n_{j = 1} g'_j (\zeta_j) \,d \zeta_j, \] whenever \(f (z) = \sum^{\infty}_{| \alpha| = 0} a_{\alpha} z^{\alpha}\) is an analytic function on \(D_n\). The authors study the boundedness of the operator \(T_{\vec g}\) on the Hardy space \(H^p (D_n)\), \(0 < p < \infty\), on the generalized weighted Bergman space \(A^{p, q}_{\mu} (D_n)\), \(0 < p, q < \infty\), and on the \(\alpha\)-Block space \(B^{\alpha} (D_n)\), \(\alpha > 1\).