Weak spectral order of Hardy, Littlewood and Pólya
type: Item
instance of: scholarly article
title: Weak spectral order of Hardy, Littlewood and Pólya
zbMATH Open document ID: 0589.46022
DOI: 10.1016/0022-247X(85)90004-6
author: Yūji Sakai
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: This paper extends the classical weak spectral order of Hardy, Littlewood and Pólya to a class of functions which includes all the finite-real valued measurable functions. Let \((X,\Lambda,\mu)\) be a measure space with \(0<\mu (X)=a\leq +\infty\), and denote by \({\mathcal M}(X)\) the set of all extended-real valued measurable functions on X. To each \(f\in {\mathcal M}(X)\) assign its decreasing rearrangement \(f^*\) on [0,a) defined by \(f^*(s)=\sup \{t:d_ f(t)>s\}\), where \(d_ f(t)\equiv \mu (\{x: f(x)>t\})\) is the distribution function of f. The author considers a new class of measurable functions \({\mathcal P}(X)=\{f\in {\mathcal M}(X):\) the value of the integral of \(f^*\) over [0,s] is definite, finitely or infinitely for all \(s\in [0,a)\}\) and shows first that \({\mathcal P}(X)=\{f\in {\mathcal M}(X):\lim_{t\to - \infty}d_ f(t)=a\) or \(f^+\in L^ 1(X)+L^{\infty}(X)\}\supset \{f\in {\mathcal M}(X):f>-\infty\) a.e. X\}. Next, he defines a preorder relation \(\ll\) (which he calles the weak spectral order of Hardy, Littlewood and Pólya) on \({\mathcal P}(X)\cup {\mathcal P}(X')\) by the requirements that \(f\ll g\) if and only if \(f\in {\mathcal P}(X)\), \(g\in {\mathcal P}(X')\) and the value of the integral of \(f^*\) over [0,s] is less than or equal to that of \(g^*\) over [0,s] for all \(s\in [0,a).\) One of the main result states: Let \((X,\Lambda,\mu)\) and \((X',\Lambda',\mu')\) be two measure spaces with \(\mu (X)=\mu'(X')\) and let \(f\in {\mathcal P}(X)\), \(g\in {\mathcal P}(X')\). Then \(f\ll g\) if and only if \[ \int_{X}(f-u)^+d\mu \leq \int_{X'}(g-u)^+d\mu ' \] for all \(u\in {\mathbb{R}}.\) Moreover he proves that increasing convex functions preserve the weak spectral order: Let \((X,\Lambda,\mu)\) and \((X',\Lambda',\mu')\) be two measure spaces with \(\mu(X)=\mu'(X')\), and let \(f\in {\mathcal P}(X)\) and \(g\in {\mathcal P}(X')\). Then (i) \(f\ll g\) if and only if \(\Phi(f)\ll \Phi(g)\) for all increasing and left-continuous convex …
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 94A15
zbMATH DE Number: 3946897
zbMATH Keywords: majorization
zbMATH Keywords: convex functions
zbMATH Keywords: weak spectral order of Hardy, Littlewood and Pólya
zbMATH Keywords: finite-real valued measurable functions
zbMATH Keywords: decreasing rearrangement
zbMATH Keywords: preorder relation
zbMATH Keywords: increasing convex functions preserve the weak spectral order
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3946897