Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1166.32002
Shamoyan, Romi; Li, Songxiao
On some properties of a differential operator on the polydisk. (English)
[J] Banach J. Math. Anal. 3, No. 1, 68-84, electronic only (2009). ISSN 1735-8787/e

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 $w= |w|\zeta$.\par From here some new embedding theorems for various quasinorms, where the operators $R^s$ are participating, are obtained.
[Sergey M. Ivashkovich (Villeneuve d'Ascq)]

MSC 2000:: *32A18 Bloch functions etc.
32A36 Bergman spaces

Keywords: holomorphic function; differential operator

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]