## Estimates of an integral operator on function spaces.(English)Zbl 1052.47044

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 $$B^{\alpha} (D_n)$$, $$\alpha > 1$$.

### MSC:

 47G10 Integral operators 47B38 Linear operators on function spaces (general) 46E20 Hilbert spaces of continuous, differentiable or analytic functions
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