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We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov-Götze characterization of measures satisfying a logarithmic Sobolev inequality. As an application of the criterion we present a soft proof of the Latała-Oleszkiewicz inequality for exponential measures, and describe the measures on the line which have the same property. New concentration inequalities for product measures follow....

Given a metric continuum $X$ and a positive integer $n$, $F_n\left(X\right)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_n\left(X\right)$, $F_n(K,X)$ denotes the set of elements of $F_n\left(X\right)$ containing $K$ and $F_n^K\left(X\right)$ denotes the quotient space obtained from $F_n\left(X\right)$ by shrinking $F_n(K,X)$ to one point set. Given a map $f:X\to Y$ between continua, $f_n:F_n\left(X\right)\to F_n\left(Y\right)$ denotes the induced map defined by $f_n\left(A\right)=f\left(A\right)$. Let $K\in F_n\left(X\right)$, we shall consider the induced map in the natural way $f_{n,K}:F_n^K\left(X\right)\to F_n^{f\left(K\right)}\left(Y\right)$. In this paper we consider the maps $f$, $f_n$, $f_{n,K}$ for some $K\in F_n\left(X\right)$ and $f_{n,K}$ for...