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Logarithmic Sobolev inequalities and contractivity properties of semigroups. (English) Zbl 0812.47037

Dell’Antonio, Gianfausto (ed.) et al., Dirichlet forms. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8-19, 1992. Berlin: Springer-Verlag. Lect. Notes Math. 1563, 54-88 (1993).
The paper studies the relationship between the logarithmic Sobolev inequalities and the contractive properties of semigroups.
After a description of quite general properties of logarithmic Sobolev inequalities over an arbitrary finite measure space, the basic relationship between logarithmic Sobolev inequalities generated by an operator \(H\) and \(L^ p\) to \(L^ q\) contraction properties of \(e^{- tH}\) are established. Three kinds of contractivity properties of \(e^{- tH}\) are studied (hyper, super and ultra contractivity) and their relation to logarithmic Sobolev inequalities.
Finally, also a survey of the contexts in which logarithmic Sobolev inequalities have been used so far is given.
For the entire collection see [Zbl 0782.00060].
Reviewer: S.Totaro (Firenze)

MSC:

47D06 One-parameter semigroups and linear evolution equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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