## An example in the theory of hypercontractive semigroups

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- by Andrzej Korzeniowski and Daniel W. Stroock PDF
- Proc. Amer. Math. Soc.
**94**(1985), 87-90 Request permission

## 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 $L$.## References

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## Additional Information

- © Copyright 1985 American Mathematical Society
- 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