New proof of the Semmes inequality for the derivative of the rational function
type: Item
instance of: scholarly article
title: New proof of the Semmes inequality for the derivative of the rational function
zbMATH Open document ID: 1130.30312
DOI: 10.1023/A:1019802112633
author: A. A. Pekarskii
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: For a function \(f\) analytic on the open unit disc \(D\) and \(\alpha> 0\), by \({\mathcal T}^\alpha f\) we denote the Riemann-Liouville derivative of order \(\alpha\). \(H_p\) \((0< p\leq \infty)\) denotes the Hardy space and \(B_p\) \((0< p\leq\infty)\) denotes the Besov space. Let \(H^\alpha_p\) be the set of \(f\) with \({\mathcal T}^\alpha f\in H_p\) and \(\| f\|_{H^\alpha_p}= \|{\mathcal T}^\alpha f\|_{H_p}\), and \(B^\alpha_p\) the set of \(f\) with \({\mathcal T}^\alpha f\in B^\alpha_p\) and \(\| f\|_{B^\alpha_p}= \|{\mathcal T}^\alpha f\|_{B_p}\). The author shows that if all the poles of the rational function \(R\) of degree \(n\), \(n= 1,2,3,\dots\), lie in \(\mathbb C\setminus D\), then \(\| R\|_{H^\alpha_{1/2}}\leq cn^\alpha\| R\|_\beta\), \(\| R\|_{\beta^\alpha_{1/2}}\leq cn^\alpha\| R\|_\beta\), where \(\alpha> 0\) and \(c> 0\) depends only on \(\alpha\). For the proof, the author uses the special integral representation of rational functions.
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1928280
zbMATH Keywords: Semmes inequality
zbMATH Keywords: rational function
zbMATH Keywords: Bloch space
zbMATH Keywords: Hardy-Sobolev space
zbMATH Keywords: Hardy-Besov space
zbMATH Keywords: inequality of Bernstein type
zbMATH Keywords: Blaschke product
reviewed by: Takahiko Nakazi
MaRDI profile type: Publication
full work available at URL: a:1019802112633
OpenAlex ID: W73513573
description: scientific article; zbMATH DE number 1928280