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Logarithmic Sobolev inequalities and exponential integrability. (English) Zbl 0846.46020

Let \((X, B, m)\) be a probability space and \(E\) a Dirichlet form on \(L^2 (X, m)\). The authors associate to \(E\) a bilinear map \(\Gamma: D(E) \times D(E)\to L^1 (X,m)\), by means of which they can define Lipschitz continuous functions.
If \(E\) satisfies a logarithmic Sobolev inequality, then the Lipschitz continuous functions are exponentially integrable. An explicit upper bound for the integral is given. The method used is due to E. B. Davies and B. Simon [J. Funct. Anal. 59, 335-395 (1984; Zbl 0568.47034)]. The result extends X. Fernique’s theorem [C. R. Acad. Sci., Paris, Sér. A 270, 1698-1699 (1970; Zbl 0206.19002)].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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