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Sobolev inequalities for probability measures on the real line. (English) Zbl 1072.60008
A probability measure \(\mu \) on \(R\) satisfies Poincaré inequality if there exists a constant \(C_{\text P}>0\) such that for every smooth function \(f:R\rightarrow R\) \[ \int f^2\,d\mu-\biggl(\int f\,d\mu\biggr)^2\leq C_{\text P}\int(f^{\prime})^2\,d\mu \] and a logarithmic Sobolev inequality if there exists a constant \(C_{\text{LS}}>0\) such that for every smooth function \(f:R\rightarrow R\) \[ \int f^2\log f^2\,d\mu-\biggl(\int f^2\,d\mu\biggr)\log \biggl(\int f^2\,d\mu\biggr)\leq C_{\text{LS}}\int (f^{\prime})^2\,d\mu. \]
The authors give lower and upper bounds for the optimal values of \(C_{\text P}\) and \(C_{\text{LS}}\). Similar results are established for certain other classes of inequalities.

MSC:
60E15 Inequalities; stochastic orderings
26D15 Inequalities for sums, series and integrals
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