Aida, Shigeki; Masuda, Takao; Shigekawa, Ichiro Logarithmic Sobolev inequalities and exponential integrability. (English) Zbl 0846.46020 J. Funct. Anal. 126, No. 1, 83-101 (1994). 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)]. Reviewer: E.De Pascale (Rende) Cited in 52 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:probability space; Dirichlet form; Lipschitz continuous function; logarithmic Sobolev inequality; exponentially integrable Citations:Zbl 0568.47034; Zbl 0206.19002 PDFBibTeX XMLCite \textit{S. Aida} et al., J. Funct. Anal. 126, No. 1, 83--101 (1994; Zbl 0846.46020) Full Text: DOI