On cones of step functions in rearrangement invariant spaces
type: Item
instance of: scholarly article
title: On cones of step functions in rearrangement invariant spaces
zbMATH Open document ID: 0828.46019
DOI: 10.1007/BF00975159
published in: Siberian Mathematical Journal
publication date: 1994-11-08T00:00:00Z
review text: The author gives conditions on a family of sets \(\{e_k\}\) and on a space \(X\) for which norm equivalence on the cone generated by \(\{\chi_{e_k}\}\) implies norm equivalence. If \(X\) is a rearrangement invariant space (r.i.) of functions on \([0, 1]\), it is known that \(X\) is uniquely determined by the cone of functions \(y\in X\) of the form \[ y(t)= \sum_{k=1}^\infty c_k \chi_{(2^{-k}, 2^{-k+ 1}]} (t), \] where \(c_k\geq 0\), \(c_k\leq c_{k+1}\). If \(\Phi\) is a class of rearrangement invariant spaces and \(X_0\in \Phi\), we say that the system \(\{ e_k \}^\infty_{k =1}\) of subsets of \([0, 1]\) determines \(X_0\) in the class \(\Phi\) if the equivalence of the norms of \(X_0\) and of an arbitrary r.i. space \(X\in \Phi\) on the cone of functions of the form \[ y= \sum_{k=1 }^\infty c_k \chi_{e_k} \qquad (c_k\geq 0,\;c_k\leq c_{k+1}) \] implies that \(X= X_0\) (isomorphism). The author works with the rearrangement invariant Lorentz space \(\Lambda (\varphi)\), and the Marcinkiewicz space, \(M(\varphi)\): \[ |x|_{\Lambda (\varphi)}= \int_0^1 x^* (s) d\varphi (s), \qquad |x|_{M (\varphi)}= \sup_{0< t\leq 1} \int_0^t x^* (s) ds/ \varphi (t), \] where \(\varphi(t)\) is an increasing concave function on \([0, 1]\), \(\varphi (0)=0\). Take a collection of sets \(\{e_k\}\), subsets of \([0, 1]\), whose measures \(\mu (e_k)= a_k\) satisfy \(0< a_{k+1}\leq a_k\), \(\sum_{k=1 }^\infty a_k\leq 1\), and \(\sup_{k=1, 2,\dots} {1\over {a_k}} \sum_{n=k}^\infty a_n< +\infty\). Let \(M\) denote the dilation function of \(\varphi\), and \(\gamma_\varphi\), \(\delta_\varphi\) the upper and lower dilation exponents of \(\varphi\), where \[ M_\varphi (t) = \sup_{0< s\leq \min (1, 1/t)} \varphi (st)/ \varphi (s),\qquad t>0, \] \[ \gamma_\varphi = \lim_{t\to 0+} \ln M_\varphi (t)/\ln (t), \qquad \delta_\varphi = \lim_{t\to \infty} \ln M_\varphi (t)/ \ln (t). \] In the cases considered by the author, \(\varphi\) can be taken as the fundamental function of the r.i. space \(X\), na…
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 652166
zbMATH Keywords: rearrangement invariant space
zbMATH Keywords: rearrangement invariant Lorentz space
zbMATH Keywords: Marcinkiewicz space
zbMATH Keywords: upper and lower dilation exponents of \(\varphi\)
zbMATH Keywords: intermediate space
author: Sergey Astashkin
reviewed by: Raymond Johnson
MaRDI profile type: Publication
cites work: Q3957448
cites work: A note on norms of compression operators on function spaces
full work available at URL: https://doi.org/10.1007/bf00975159
OpenAlex ID: W2325938039
description: scientific article; zbMATH DE number 652166