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A Family of Integral Inequalities on the Interval \([-1,1]\)

  • Ali Hafidi
  • Moulay Rchid Sidi Ammi
  • Praveen Agarwal
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We study the heat semigroup \((P^{n}_{t})_{t\ge 0}=\{\exp (tL_{n})\}_{t\ge 0}\) generated by the Gegenbauer operator \(L_{n}:=(1-x^{2})\frac{d^{2}}{dx^{2}}-nx\frac{d}{dx}\), on the interval \([-1,1]\) equipped with the probability measure    \(\mu _{n}(dx):=c_{n}(1-x^{2})^{\frac{n}{2}-1}\), where \(c_{n}\) the normalization constant and n is a strictly positive real number. By means of a simple method involving essentially a commutation property between the semigroup and derivation, we describe a large family of optimal integral inequalities with logarithmic Sobolev and Poincaré inequalities as particular cases.

Keywords

Heat semigroup Gegenbauer operator Spectral gap Poincaré’s inequality Sobolev’s inequality Logarithmic Sobolev inequality \(\varphi \)-entropy inequality 

Mathematics Subject Classification 2010

39B62 39B72 44A15 46E35 60J25 

References

  1. 1.
    A. Arnold, J.-P. Bartier, J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré iequalities. Commun. Math. Sci. 5(4), 971–979 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Bentaleb, S. Fahlaoui, A. Hafidi, Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup. Semigroup Forum 85(2), 361–368 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Bentaleb, S. Fahlaoui, A family integral inequalities on the circle \(S^{1}\). Proc. Jpn. Acad. Ser. A 86, 55–59 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Boutahir, A. Hafidi, Family of functional inequalities for the uniform measure. J. Math. Sci. Appl. 5(1), 19–23 (2017)Google Scholar
  5. 5.
    L. Gross, Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and Its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)CrossRefGoogle Scholar
  7. 7.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)CrossRefGoogle Scholar
  8. 8.
    F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites. Potential Anal. 22(1), 1–15 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Ali Hafidi
    • 1
  • Moulay Rchid Sidi Ammi
    • 1
  • Praveen Agarwal
    • 2
    • 3
  1. 1.Department of Mathematics, AMNEA Group, Faculty of Sciences and TechniquesMoulay Ismail UniversityErrachidiaMorocco
  2. 2.Department of MathematicsAnand International College of EngineeringJaipurIndia
  3. 3.International Centre for Basic and Applied SciencesJaipurIndia

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