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This article is cited in 4 scientific papers (total in 4 papers)
New Proof of the Semmes Inequality for the Derivative of the Rational Function
A. A. Pekarskii Yanka Kupala State University of Grodno
Abstract:
In the open disk $|z|<1$ of the complex plane, we consider the following spaces of functions: the Bloch space $\mathscr B$; the Hardy–Sobolev space $H^\alpha _p$, $\alpha \ge 0$, $0<p\le \infty $; and the Hardy–Besov space $B^\alpha _p$, $\alpha \ge 0$, $0<p\le \infty $. It is shown that if all the poles of the rational function $R$ of degree $n$, $n=1,2,3,\dots $, lie in the domain $|z|>1$, then $\|R\|_{H^\alpha _{1/\alpha }}\le cn^\alpha \|R\|_{\mathscr B}$,
$\|R\|_{B^\alpha _{1/\alpha }}\le cn^\alpha \|R\|_{\mathscr B}$,
where $\alpha >0$ and $c >0$ depends only on $\alpha$ . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.
Received: 10.09.1998
Citation:
A. A. Pekarskii, “New Proof of the Semmes Inequality for the Derivative of the Rational Function”, Mat. Zametki, 72:2 (2002), 258–264; Math. Notes, 72:2 (2002), 230–236
Linking options:
https://www.mathnet.ru/eng/mzm419https://doi.org/10.4213/mzm419 https://www.mathnet.ru/eng/mzm/v72/i2/p258
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Abstract page: | 496 | Full-text PDF : | 196 | References: | 47 | First page: | 1 |
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