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Integral inequalities related to the Tchebychev semigroup

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Abstract

We study the semigroup (P t ) t≥0 generated by the operator

$$\mathcal{L}:=(1-x^{2}){\frac{d^{2}}{dx^{2}}}-x\,{\frac{d}{dx}}$$

acting on the space \(\mathbb{L}^{2}([-1,+1],\sigma)\) with respect to the probability measure \(\sigma (dx):={\frac{1}{\pi \sqrt{1-x^{2}}}}\ dx.\) We prove the Sobolev and Onofri inequalities by means of an elementary method involving essentially a commutation property between the semigroup and the derivation.

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Authors and Affiliations

  1. Faculty of Sciences, Department of Mathematics and Informatics, University Moulay Ismaïl, BP 11201, Zitoune, Meknès, Morocco

    Abdellatif Bentaleb & Said Fahlaoui

Authors
  1. Abdellatif Bentaleb
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  2. Said Fahlaoui
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Correspondence to Abdellatif Bentaleb.

Additional information

Communicated by Jerome A. Goldstein.

This paper was written while the first author was visiting the International Centre for Theoretical Physics, Trieste (Italy) in June 2007.

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Bentaleb, A., Fahlaoui, S. Integral inequalities related to the Tchebychev semigroup. Semigroup Forum 79, 473–479 (2009). https://doi.org/10.1007/s00233-009-9160-2

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  • DOI: https://doi.org/10.1007/s00233-009-9160-2

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