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On the minimization of Sobolev constants and logarithmic Sobolev constants for Jacobi and Laguerre operators. (Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre.) (French) Zbl 0914.60047

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 14-29 (1998).
Summary: This paper concerns Sobolev constants for Jacobi operators and logarithmic Sobolev constants for Laguerre operators. These constants are different from the first non-negative eigenvalue associated to these operators. We suggest a method based on the existence of extremal functions and on the study of the sign of second degree polynomials to bound these constants. These new lower bounds allow us to recover, on one hand, the Sobolev constant due to Bakry for optimal exponent and, on the other hand, the logarithmic Sobolev constant obtained by Saloff-Coste.
For the entire collection see [Zbl 0893.00035].

MSC:

60J45 Probabilistic potential theory
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