Abstract

We consider on the interval [-1,1] the heat semigroup generated by the Legendre operator acting on the Hilbert space with respect to the uniform measure By means of a simple method involving some semigroup techniques, we describe a large family of optimal integral inequalities with the Poincaré and logarithmic Sobolev inequalities as particular cases.

Highlights

  • Gross’ logarithmic Sobolev inequality [2] states that for all smooth functions f on d ( ) ( ) ∫ ∫ ∫ d f 2 log f 2dγ d − d f 2dγ d log d f dγ d ( ) ∫ ≤ 2 d ∇f 2dγ d, (1)white dγ d denote the normalized Gaussian measure on ( ) = d : dγ d ( x) 2π −d exp − x 2 / 2

  • In 1989, W.Bekner [2] derived a family of generalized Poincaré inequalities that yield a sharp interpolation between Poincaré inequality and logarithmic Sobolev inequality: ( ) ∫ ∫ d f 2dγ d − d etL f dγ d

  • A.Bentaleb, S.Fahlaoui and A.Hafidi proposed in [[3], Section 2] a generalized of the inequality 3 and obtained the following inequality: for all smooth function f on d

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Summary

Introduction

Gross’ logarithmic Sobolev inequality [2] states that for all smooth functions f on d ( ) ( ) ∫ ∫ ∫ d f 2 log f 2dγ d − d f 2dγ d log d f dγ d. White dγ d denote the normalized Gaussian measure on ( ) = d : dγ d ( x) 2π −d exp − x 2 / 2. In this Gaussian context, the Poincaré inequality(spectral gap inequality) is given by:. In 1989, W.Bekner [2] derived a family of generalized Poincaré inequalities that yield a sharp interpolation between Poincaré inequality and logarithmic Sobolev inequality:. The purpose of this paper is to present a family of integral inequalities on the interval [-1,1] which provide interpolation between the Sobolev and Poincaré inequalities (see Theorem 1 below)

Preliminaries
The Main Result

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