Distributional control for operators on vector-valued \(L^p\)-spaces
type: Item
instance of: scholarly article
title: Distributional control for operators on vector-valued \(L^p\)-spaces
zbMATH Open document ID: 0907.47024
author: Brian P. Kelly
author: Nakhlé H. Asmar
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-03-02T00:00:00Z
full work available at URL: CONT27-1.html
review text: Let \((\Omega,{\mathcal J},\mu)\) be a \(\sigma\)-finite measure space, and let \((X,\|\cdot\|)\) be a Banach space. The space \(L^p(\Omega,\mu,X)\) consisting of \(X\)-valued \(p\)th power integrable functions on \(\Omega\) is denoted for convenience by \(E^p\). The distribution \(\phi(f;\cdot)\) of a function \(f:\Omega\to X\) is defined by \(\phi(f; y)= \phi(N(f);y)= \mu(\{w\in \Omega:\| f(w)\|> y\})\), \(y>0\). An operator \(T: E^1\cap E^\infty\to E^1\cap E^\infty\) is said to be `\(\mu\)-distributionally controlled' if there are positive constants \(c\), \(\gamma\), such that, for \(y>0\), \(f\), \(g\) in \(E^1\cap E^\infty\), (i) \(\phi(T(f);y)\leq c\phi(f;\gamma y)\) and (ii) \(\phi(\min\{\| T(f)\|, \| T(g)\|\}; y)\leq c\phi(\min\{\| f\|, \| g\|\};\gamma y)\). If \(G\) is a locally compact Abelian group, then a representation \(u\to R_u\) of \(G\) is said to be \(\mu\)-distributionally controlled if (i) and (ii) are valid for \(T= R_u\), \(u\in G\). In the first main theorem of this paper, the authors show that if \(T\) is \(\mu\)-distributionally controlled, with constants \(c\), \(\gamma\) as in (i), (ii), then for \(1\leq p<\infty\), there is a linear operator \(T^{(p)}\) on \(E^p\) such that \(T^{(p)}(f)= T(f)\) for \(f\in E^1\cap E^\infty\) and \(\| T^{(p)}\|\leq c^{1/p}\gamma^{- 1}\). In another theorem, it is shown that if \(u\to S_u\) is a representations of \(G\) on \(E^p\), then there are \(\mu\)-distributionally controlled representations of \(G\) such that \(S_u= R^{(p)}_u\), \(u\in G\), if and only if each operator \(S_u\) is separation preserving and bounded in \(E^p\), \(1\leq p<\infty\), and on \(E^p\cap E^\infty\) with \(L^\infty\)-norm. In a result relating to bounded representations, it is shown that if \(u\to R_u\) is a representation of \(G\) which is uniformly bounded on \(E^p\) for \(p= p_1\) or \(p= p_2\), \(1\leq p_1\), \(p_2<\infty\), then the representation is strongly continuous on \(E^{p_1}\) if and only if it is strongly continuo…
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1051525
zbMATH Keywords: operators on vector-valued \(L^p\)-spaces
zbMATH Keywords: distribution functions
zbMATH Keywords: vector-valued operators
zbMATH Keywords: representations of a group
zbMATH Keywords: \(\mu\)-distributionally controlled representation
reviewed by: George O. Okikiolu
MaRDI profile type: Publication
OpenAlex ID: W2075824449
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DOI: 10.1216/RMJM/1181071950
description: scientific article; zbMATH DE number 1051525