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Mat. Zametki, 2002, Volume 72, Issue 2, Pages 258–264 (Mi mz419)  

This article is cited in 2 scientific papers (total in 2 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 $n$, $n=1,2,3,…$, 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.

DOI: https://doi.org/10.4213/mzm419

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English version:
Mathematical Notes, 2002, 72:2, 230–236

Bibliographic databases:

UDC: 517.53
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

Citation in format AMSBIB
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\paper New Proof of the Semmes Inequality for the Derivative of the Rational Function
\jour Mat. Zametki
\yr 2002
\vol 72
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\pages 258--264
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\jour Math. Notes
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\pages 230--236
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. F. Shamoyan, “Kharakterizatsii tipa VMO, diagonalnoe otobrazhenie i ogranichennnost integralnykh operatorov v nekotorykh prostranstvakh analiticheskikh funktsii”, Vladikavk. matem. zhurn., 9:2 (2007), 40–53  mathnet  mathscinet
    2. Baranov A., Zarouf R., “The Differentiation Operator From Model Spaces to Bergman Spaces and Peller Type Inequalities”, J. Anal. Math., 137:1 (2019), 189–209  crossref  isi
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