Some properties of the relative rearrangement
type: Item
instance of: scholarly article
title: Some properties of the relative rearrangement
zbMATH Open document ID: 0686.28003
DOI: 10.1016/0022-247X(88)90169-2
author: J.-M. Rakotoson
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Omega \subset R^ N\) be a bounded Lebesgue measurable set of measure \(| \Omega |\). The decreasing rearrangement of a real measurable function u on \(\Omega\) is defined by \[ u_*(s)=\inf \{\theta \in R,| \{x,u(x)>\theta \}| \leq s\},\quad s\in {\bar \Omega}^*=[0,| \Omega |]. \] Given \(t\in R\) we put \(P(t)=\{x\in \Omega,\quad u(x)=t\}.\) If \(| P(t)| >0,\) then P(t) is called the flat region of value t of the function u. \textit{J. Mossino} and \textit{R. Temam} [Duke Math. J. 48, 475-495 (1981; Zbl 0476.35031)] introduced the notion of relative rearrangement: For \(v\in L^ p(\Omega),\quad 1\leq p\leq \infty,\) define a function w on \({\bar \Omega}{}^*\) by \[ w(s)=\int_{u>u_*(s)}v(x)dx+\int^{s-| \{u>u_*(s)\}|}_{0}(v|_{P(u_*(s))})_*(\sigma)d\sigma, \] where \(v|_ E\) is the restriction of v to the set E. The function \(v_{*u}=dw/ds\) satisfies \(\| v_{*u}\|_{L^ p(\Omega^*)}\leq \| v\|_{L^ p(\Omega)}\) and is called the relative rearrangement of v with respect to u. The author proves various properties of the relative rearrangement, among others the following generalization of the well-known Hardy-Littlewood inequality: If u is a real measurable function on \(\Omega\), \(v_ 1\in L^ p(\Omega),\quad v_ 2\in L^ q(\Omega^*),\quad 1/p+1/q=1,\) then \[ \int_{\Omega^*}v_{1*u}v_ 2 d\sigma \leq \int_{\Omega^*}v_{1*}v_{2*} d\sigma. \]
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4124024
zbMATH Keywords: decreasing rearrangement
zbMATH Keywords: relative rearrangement
zbMATH Keywords: Hardy-Littlewood inequality
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124024