Weak spectral order of Hardy, Littlewood and Pólya
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 functions \(\Phi: J\to {\bar {\mathbb{R}}}\) such that the interval J contains the ranges of f and g, provided that \(\Phi(f)\in {\mathcal P}(X)\) and \(\Phi(g)\in {\mathcal P}(X'),\) (ii) If, in addition, \(\mu(X)<\infty\) and \(\mu(\{f=\infty\})= \mu'(\{g=\infty\})=0\), then the restriction that \(\Phi\) be left- continuous may be removed. Further, he shows that \(| f^*| \ll | f|\) for any \(f\in {\mathcal M}(X).\) Also the following results are shown assuming that \((X,\Lambda,\mu)\) is \(\sigma\)-finite: \(f+g\ll f^*+g^*\) for any f,g\(\in {\mathcal M}(X)\) such that \(f^-,g^-\in L(X)+L^{\infty}(X)\), and \(f\ll g\) if and only if \(fh\ll g^*h^*\) for any \(h\in {\mathcal M}^+(X)\) provided that \(f,g\in {\mathcal M}^+(X)\).
weak spectral order of Hardy, Littlewood and Pólya
increasing convex functions preserve the weak spectral order
scientific article; zbMATH DE number 3946897