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Hilbert operator on the polydisk. (English) Zbl 1088.47024
Let \(H^p=H^p(\mathbb U^n)(0<p<\infty)\) denote the Hardy space defined on the polydisk \(\mathbb U^n\), that is,
\[ H^p=\{ f: f \text{ is a holomorphic function on } \mathbb U^n \text{and} \parallel f \parallel_p<\infty\}, \] where
\[ \parallel f \parallel_p=\left\{ \sup_{0\leq r<1}\int_{[ 0,2\pi]}{}^n | f(r\cdot e^{i\theta})|^pd\theta/(2\pi)^n\right\}^{1/p} \]
and \(r\cdot e^{i\theta}=(r_1e^{i\theta 1}, \dots, r_ne^{i\theta n})\), \(d\theta=d\theta_1\dots d\theta_n\). If \[ f(z_1,\dots, z_n)= \displaystyle\sum^\infty_{k^1+\dots+k^n=0} a_{k^1,\dots,k^n} z_1^{k^1}\dots {z_n}^{k^n} \] is a holomorphic function defined on \(\mathbb U^n\), the Hilbert type operator \(H_n\) is defined by \[ H_n(f)(z_1, \dots, z_n)=\displaystyle \sum^\infty_{k^1+\dots+k^n=0}\left\{\displaystyle\sum^1_{i^1,\dots, i^n\geq 0}[ a_{i^1,\dots,i^n}/\prod_{j=1,\dots,n}(k_j+i_j+1)]\right\} z_1^{k^1}\dots z_n^{k^n}. \] In the paper under review, the author proves the following result:
(1) If \(2\leq p<\infty\), then \[ \parallel H_n(f) \parallel_p\leq [\pi/sin(\pi/p)]^n\parallel f \parallel_p \] for each \(f\in H^p\).
(2) If \(0<p<2\), then
\[ \parallel H_n(f)\parallel_p\leq[\pi/sin(\pi/p)]^n\parallel f\parallel_p \] for each \(f\in H^p\) such that \(f(z_1, \dots, z_n)=z_1\dots z_n f_0(z_1, \dots, z_n)\) for some \(f_0\in H^p\).

MSC:
47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
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