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

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

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
\Bibitem{Pek02} \by A.~A.~Pekarskii \paper New Proof of the Semmes Inequality for the Derivative of the Rational Function \jour Mat. Zametki \yr 2002 \vol 72 \issue 2 \pages 258--264 \mathnet{http://mi.mathnet.ru/mz419} \crossref{https://doi.org/10.4213/mzm419} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1942550} \zmath{https://zbmath.org/?q=an:1130.30312} \transl \jour Math. Notes \yr 2002 \vol 72 \issue 2 \pages 230--236 \crossref{https://doi.org/10.1023/A:1019802112633} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000178299100024} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0141513946} 

• http://mi.mathnet.ru/eng/mz419
• https://doi.org/10.4213/mzm419
• http://mi.mathnet.ru/eng/mz/v72/i2/p258

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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
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
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