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Boundedness and compactness of an integral operator on a weighted space on the polydisc. (English) Zbl 1121.47032
The author considers the boundedness and compactness of the integral operator \[ T_g(f)=\int_0^{z_1}\dots\int_0^{z_n} f(\xi_1,\dots,\xi_n)\,g(\xi_1,\dots,\xi_n) \,d\xi_1\dots d\xi_n \] on the space of analytic functions on the polydisk \(U^n\) such that \[ \sup_{z\in U^n} \prod_{i=1}^n(1-| z_i| )^{\alpha_i} | f(z_1,\dots,z_n)| <\infty \] for a given \((\alpha_1,\dots,\alpha_n)\) with \(\alpha_i>0\) for all \(i=1,\dots,n\). The results establish that the boundedness is characterized by \(\prod_{i=1}^n(1-| z_i|)| g(z_1,\dots,z_n)|<\infty\) and the compactness the corresponding “little o” condition. Moreover, a criterion for the membership of a function in the \(p\)-Bloch space on the polydisc is obtained, which extends the result of F. Holland and D. Walsh [Math. Ann. 273, 317–335 (1986; Zbl 0561.30025)].

47G10 Integral operators
47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
32A36 Bergman spaces of functions in several complex variables
32A70 Functional analysis techniques applied to functions of several complex variables