Differentiation of Banach-space-valued additive processes
type: Item
instance of: scholarly article
title: Differentiation of Banach-space-valued additive processes
zbMATH Open document ID: 0987.47004
author: Ryotaro Sato
published in: Studia Mathematica
publication date: 2001-12-11T00:00:00Z
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 1681003
zbMATH Keywords: vector-valued local ergodic theorem and differentiation theorem
zbMATH Keywords: reflexive Banach space
zbMATH Keywords: \(d\)-dimensional semigroup of linear contractions
zbMATH Keywords: contraction majorant
zbMATH Keywords: vector-valued Lorentz spaces and Orlicz spaces
MaRDI profile type: Publication
full work available at URL: sm147-2-3
OpenAlex ID: W2059211715
DOI: 10.4064/SM147-2-3
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review text: Let \((X,\|\cdot\|)\) be a Banach space and \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite measure space. Let \((L,\|\cdot\|_L)\) denote a Banach space of \(X\) valued strongly measurable functions on \((\Omega,\Sigma,\mu)\). \(\|\cdot\|_L\) satisfies NEWLINENEWLINENEWLINEIf \(f,g\in L\) and \(\|f(\omega)\|\leq\|g(\omega)\|\) for almost all \(\omega\in\Omega\), then \(\|f\|_L\leq\|g\|_L\). NEWLINENEWLINENEWLINEIf \(g\) is an \(X\) valued strongly measurable function on \(\Omega\) and if there exists an \(f\in L\) such that \(\|g(\omega)\|\leq\|f(\omega)\|\) for almost all \(\omega\), then \(g\in L\). NEWLINENEWLINENEWLINEIf \(E_n\in\Sigma\), \(E_n\supset E_{n+1}\) for each \(n\geq 1\), and \(\bigcap_{n=1}^\infty E_n=\emptyset\), then for any \(f\in L\) \(\lim_{n\to\infty}\|\chi_{E_n}f\|_L=0\), where \(\chi_E\) is the characteristic function of \(E_n\) \(f,g\in L\), \(\|f(\omega)\|\leq\|g(\omega)\|\) a.e. and \(\|f\|_L=\|g\|_L\), then \(\|f(\omega)=\|g(\omega)\|\) a.e. NEWLINE\[NEWLINE\{f_n\}\subset LNEWLINE\]NEWLINE such that \(\|f_n(\omega)\|\leq\|f_{n+1}(\omega)\|<\infty\), then there exists \(f\in L\) such that \(\|f_n(\omega)\|\leq\|f(\omega)\|\) a.e. Fix \(x_1\in X\) such that \(\|x_1\|=1\). \(L(R)\) is the set of real valued measurable functions \(\widetilde f\) on \((\Omega,\Sigma,\mu)\) such that \(f(\omega)=\widetilde f(\omega)x_1\in L\), and define \(\|\widetilde f\|_{L(R)}=\|f\|_L\). A positive operator \(P\) on \(L(R)\) is called a majorant of a linear operator \(U\) on \(L\) if \(\|Uf(\omega)\|\leq[P\|f(\cdot)\|](\omega)\) a.e.NEWLINENEWLINENEWLINELet \(P_d=\{u=(u_1,\ldots,u_d): u_i>0, 1\leq i\leq d\}\) and \({\mathcal I}_d\) denotes the set of all the bounded intervals in \(P_d\).NEWLINENEWLINENEWLINEConsider a strongly continuous \(d\)-dimensional semigroup \(T=\{T(u): u\in P_d\}\). A set function \(F:{\mathcal I}_d\to L\) is called a bounded process if NEWLINE\[NEWLINE\sup\left\{{\|F(I)_L\over\lambda_d(I)}: I\in{\mathcal I}_d, \lambda_d(I)>0\right\}<…
reviewed by: Makoto Mori
description: scientific article; zbMATH DE number 1681003