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
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A. Arnold, J.-P. Bartier, J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré iequalities. Commun. Math. Sci. 5(4), 971–979 (2007)
A. Bentaleb, S. Fahlaoui, A. Hafidi, Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup. Semigroup Forum 85(2), 361–368 (2012)
A. Bentaleb, S. Fahlaoui, A family integral inequalities on the circle \(S^{1}\). Proc. Jpn. Acad. Ser. A 86, 55–59 (2010)
K. Boutahir, A. Hafidi, Family of functional inequalities for the uniform measure. J. Math. Sci. Appl. 5(1), 19–23 (2017)
L. Gross, Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)
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)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)
F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites. Potential Anal. 22(1), 1–15 (2005)
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Hafidi, A., Ammi, M.R.S., Agarwal, P. (2018). A Family of Integral Inequalities on the Interval \([-1,1]\). In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_17
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DOI: https://doi.org/10.1007/978-981-13-3013-1_17
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