A family of integral inequalities on the circle \(S^{1}\)
type: Item
instance of: scholarly article
title: A family of integral inequalities on the circle \(S^{1}\)
zbMATH Open document ID: 1204.47046
author: Abdellatif Bentaleb
author: S. Fahlaoui
published in: Proceedings of the Japan Academy. Series A
publication date: 2010-08-20T00:00:00Z
review text: The purpose of this paper is to present a family of integral inequalities on the unit circle \(S^1\) which provide interpolation between the Sobolev, the logarithmic Sobolev (of C.\,E.\thinspace Mueller and F.\,B.\thinspace Weissler) and Poincaré inequalities. These types of integral inequalities are deeply related to the aspects of the large-time behavior of parabolic PDE's [cf.\ \textit{A.\,Pazy}, ``Semigroups of linear operators and applications to partial differential equations'' (Applied Mathematical Sciences 44; New York etc.:\ Springer-Verlag) (1983; Zbl 0516.47023)]). One of the applications of the paper's main result is a hypercontractive estimation for the Chebyshev semigroup of operators \((P_t)_{t>0}\) generated by the Chebyshev operator \(\mathfrak{L}\) defined by \[ \mathfrak{L}:=(1-x^2)\frac{d^2}{dx^2}-x\frac{d}{dx},\quad x\in I=[-1,1] \] and acting on \(L^2(I,\mu)\) with respect to the probability measure \[ \mu(dx):=\frac{1}{\pi\sqrt{1-x^2}}dx. \]
reviewed by: Roman Wituła
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 42A99
zbMATH DE Number: 5773630
zbMATH Keywords: Chebyshev semigroup
zbMATH Keywords: spectral gap
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: logarithmic Sobolev inequality
MaRDI profile type: Publication
full work available at URL: pjaa.86.55
OpenAlex ID: W2019089782
cites work: Interpolation between logarithmic Sobolev and Poincaré inequalities
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cites work: Q4725163
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cites work: A generalization of Poincaré and log-Sobolev inequalities
DOI: 10.3792/PJAA.86.55
description: scientific article; zbMATH DE number 5773630