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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
##### Keywords:
concentration; Poincaré inequalities; optimal constants
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