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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
##### Keywords:
Hardy space; polydisk; Hilbert type operator