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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 $\Delta^n$: $$\align R^sf &= \sum_{h_1,\dots, h_n\ge 0} (k_1+\cdots+ k_n+ 1)^s a_{k_1\cdots k_n} z^{k_1}_1\cdots y^{k_n}_n,\\ D^\alpha f &= \sum_{k_1,\dots, k_n\ge 0} (k_1+ 1)^\alpha\cdots(k_n+ 1)^\alpha a_{k_1\cdots k_n} z^{k_1}_1\cdots z^k_n.\endalign$$ Here, $f(z)= \sum_{k_1,\dots, k+n\ge 0} a_{k_1\cdots k_n} z^{k_1}_1\cdots z^{k_n}_n$ belongs to $H(\Delta^n)$ or $H^p(\Delta^n)$ for some $0< p\le\infty$, $s\in\bbfR$, $\alpha\in\bbfR$. 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 $0< p<\infty$, $\alpha>-1$, $s\in\bbfN$ and $f\in H(\Delta^n)$. If $\gamma> {\alpha+2\over p}- 2$ for $p\le 1$ 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 $$\int_{\Delta^n} |D^\gamma f(z)|^p(1- |z|^2)^\alpha\, dm_{2n}(z)\le C\int^1_0 \int_{(\partial\Delta)^n} |R^sf(w)|^p(1- |w|^2)^v \,dm_n(\zeta)\,d|w|,$$ where $w= |w|\zeta$. From here some new embedding theorems for various quasinorms, where the operators $R^s$ are participating, are obtained.

32A18Bloch functions, normal functions
32A36Bergman spaces
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