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On some properties of a differential operator on the polydisk. (English) Zbl 1166.32002

The paper is devoted to the study of relations between the following differential operators in the space \(H(\Delta^n)\), \(H^p(\Delta^n)\) of holomorphic functions in the unit polydisc \(\Delta^n\):
\[ \begin{aligned} R^sf &= \sum_{h_1,\dots, h_n\geq 0} (k_1+\cdots+ k_n+ 1)^s a_{k_1\cdots k_n} z^{k_1}_1\cdots y^{k_n}_n,\\ D^\alpha f &= \sum_{k_1,\dots, k_n\geq 0} (k_1+ 1)^\alpha\cdots(k_n+ 1)^\alpha a_{k_1\cdots k_n} z^{k_1}_1\cdots z^k_n.\end{aligned} \]
Here, \(f(z)= \sum_{k_1,\dots, k+n\geq 0} a_{k_1\cdots k_n} z^{k_1}_1\cdots z^{k_n}_n\) belongs to \(H(\Delta^n)\) or \(H^p(\Delta^n)\) for some \(0< p\leq\infty\), \(s\in\mathbb{R}\), \(\alpha\in\mathbb{R}\). The authors goal is to reduce the study of \(R^s\) to a study of \(D^\alpha\), which were studied by many authors. An example of a result is the following.
Theorem 2.7. Let \(0< p<\infty\), \(\alpha>-1\), \(s\in\mathbb{N}\) and \(f\in H(\Delta^n)\). If \(\gamma> {\alpha+2\over p}- 2\) for \(p\leq 1\) or \(\gamma>{\alpha+ 1\over p}+{1\over n}(1-{1\over p})\) for \(p> 1\) and \(v= sp+\alpha n-\gamma pn+ n-1\), then
\[ \int_{\Delta^n} |D^\gamma f(z)|^p(1- |z|^2)^\alpha\, dm_{2n}(z)\leq C\int^1_0 \int_{(\partial\Delta)^n} |R^sf(w)|^p(1- |w|^2)^v \,dm_n(\zeta)\,d|w|, \]
where \(w= |w|\zeta\).
From here some new embedding theorems for various quasinorms, where the operators \(R^s\) are participating, are obtained.

MSC:

32A18 Bloch functions, normal functions of several complex variables
32A36 Bergman spaces of functions in several complex variables
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