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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An example in the theory of hypercontractive semigroups


Authors: Andrzej Korzeniowski and Daniel W. Stroock
Journal: Proc. Amer. Math. Soc. 94 (1985), 87-90
MSC: Primary 47D05; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1985-0781062-0
MathSciNet review: 781062
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Abstract: Let $L = x({d^2}/d{x^2}) + (1 - x)(d/dx)$ on ${C_c}((0,\infty ))$ be the Laguerre operator. It is shown that for $t > 0$, and $1 < p < q < \infty ,\;{e^{tl}}:{L^p}({e^{ - x}}dx) \to {L^q}({e^{ - x}}dx)$ has norm 1 if and only if ${e^{ - t}} \leqslant (p - 1)/(q - 1)$ and the corresponding logarithmic Sobolev constant is not equal to $2/\lambda$, where $\lambda$ is the smallest nonzero eigenvalue of $L$.


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Keywords: Laguerre semigroup, logarithmic Sobolev inequality, Ornstein-Uhlenbeck semigroup, hypercontractivity
Article copyright: © Copyright 1985 American Mathematical Society