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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
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

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