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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Logarithmic Sobolev inequalities for the heat-diffusion semigroup


Author: Fred B. Weissler
Journal: Trans. Amer. Math. Soc. 237 (1978), 255-269
MSC: Primary 47D05; Secondary 46E35
MathSciNet review: 479373
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Abstract: An explicit formula relating the Hermite semigroup $ {e^{ - tH}}$ on R with Gauss measure and the heat-diffusion semigroup $ {e^{t\Delta }}$ on R with Lebesgue measure is proved. From this formula it follows that Nelson's hypercontractive estimates for $ {e^{ - tH}}$ are equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map $ {L^q}(R)$ into $ {L^p}(R),1 < q < p < \infty $. Furthermore,the inequality

$\displaystyle \frac{d}{{dq}}\log \left\Vert \phi \right\Vert _q^q \leqslant \fr... ...\Vert \phi \right\Vert _q^q}}} \right] + \log {\left\Vert \phi \right\Vert _q},$

where $ 1 < q < \infty ,{J^q}\phi = (\operatorname{sgn} \phi )\vert\phi {\vert^{q - 1}}$, and the norms and sesquilinear form $ \langle ,\rangle $ are taken with respect to Lebesgue measure on $ {R^n}$, is shown to be equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map from $ {L^q}({R^n})$ into $ {L^p}({R^n})$. This inequality is analogous to Gross' logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0479373-2
PII: S 0002-9947(1978)0479373-2
Keywords: Logarithmic Sobolev inequalities, heat-diffusion semigroup, Hermite semigroup, hypercontractivity
Article copyright: © Copyright 1978 American Mathematical Society