A remark on hypercontractivity and the concentration of measure phenomenon in a compact Riemannian manifold
The concentration function \(\alpha\) (X;\(\epsilon\)), \(\epsilon >0\), of a metric space (X,d) with respect to a probability measure has been introduced by \textit{M. Gromov} and \textit{V. D. Milman} [Am. J. Math. 105, 843-854 (1983; Zbl 0522.53039)]. A geometric invariant of the Laplace operator on a compact Riemannian manifold is the hypercontractive constant \(\rho_ 0(X)\); see \textit{O. S. Rothaus} [J. Funct. Anal. 42, 102-109 (1981; Zbl 0471.58027)]. Using the method of M. Gromov and V. D. Milman, the author obtains the following estimation of \(\alpha\) (X;\(\epsilon\)) in terms of \(\rho_ 0(X)\) \[ \alpha (X;\epsilon)\leq C \exp (-c\epsilon \sqrt{\rho_ 0(X)}\log (1+\epsilon \sqrt{\rho_ 0(X)}) \] for all \(\epsilon >0\), where \(C,c>0\) are numerical constants. Some examples are given.
scientific article; zbMATH DE number 4169600
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