A class of monotone decreasing rearrangements
Let \(\mu\) be a Borel measure on \({\mathbb{R}}\) with no pure point support. The monotone decreasing rearrangement of a Borel measurable function f: \({\mathbb{R}}\to {\mathbb{R}}\) with respect to \(\mu\) is defined by \[ f^*(x)=\inf \{y\in {\mathbb{R}}: \mu_ f(y)\leq \mu (-\infty,x)\},\quad x\in {\mathbb{R}}, \] where \(\mu_ f(y)=\mu \{u\in {\mathbb{R}}: f(u)>y\}\). It is easy to see that \(f^*\) is a decreasing, right-continuous function on \({\mathbb{R}}\), and that f and \(f^*\) are equidistributed, i.e. \(\mu_ f(x)=\mu_{f^*}(x)\) for all \(x\in X\). The study of monotone decreasing rearrangements was initiated by Hardy and Littlewood in the process of treatment of fractional integrals. The notion also appears in various problems of probability theory One of the main goals of the paper under review is to prove the following rearrangement inequality for the derivative. Theorem. Let F be an increasing convex function on \([0,\infty)\), with \(F(0)=0\), and let \(\mu\) be an absolutely continuous measure on \({\mathbb{R}}\) with a strictly positive, symmetric decreasing, log-concave density. Then for every Lipschitz continuous function f and every Borel set \(A\subseteq {\mathbb{R}}\), \[ (1)\quad \int_{f^{-1}(A)}F(| {\mathcal D}f|)d\mu \geq \int_{f^{*^{-1}}(A)}F(| {\mathcal D}f^*|)d\mu. \] A similar inequality for the Lebesgue measure on an interval has been obtained by \textit{G. F. D. Duff} [Can. J. Math. 22, 408-430 (1970; Zbl 0194.360)]. The theorem above widens considerably the class of measures for which inequality (1) holds, involving as a subclass the very important class of Gaussian measures.
monotone decreasing rearrangement of a Borel measurable function
rearrangement inequality for the derivative
scientific article; zbMATH DE number 4175301
scientific article