The generalized Libera transform on Hardy, Bergman and Bloch spaces on the unit polydisc (Q1866648)

Let \(U^1= U\) be the unit disk in the complex plane \(\mathbb C\), \(dm(z)= {1\over \pi} \,dr\,d\theta\) the normalized Lebesgue measure on \(U\), \(U^n\) the unit polydisk in the complex vector space \(\mathbb C^n\) and let \(H(U^n)\) be the space of all analytic functions on \(U^n\). For \(z^0\in \overline U^n\) fixed and \(f\in H(U^n)\), the linear operator \(f\to \Lambda_{z^0}(f)\) defined by \[ \Lambda_{z^0} f(z)= {1\over \prod^n_{j=1} (z_j- z^0_j)} \int^{z_1}_{z^0_1}\cdots \int^{z_n}_{z^0_n} f(t)\,dt\quad (z\in U^n) \] is called the generalized libera transform. In this paper, the author investigates the boundedness of the generalized libera transform on Hardy, Bergman and \(a\)-Bloch spaces of analytic functions on \(U^n\).