An ergodic maximal equality for nonsingular flows and applications
type: Item
instance of: scholarly article
title: An ergodic maximal equality for nonsingular flows and applications
zbMATH Open document ID: 0813.28006
DOI: 10.1007/BF01872096
published in: Acta Mathematica Hungarica
publication date: 1995-05-28T00:00:00Z
review text: Let \(\{T_ t,\;t\in \mathbb{R}\}\) be a measurable flow of nonsingular transformations of a probability space \((X,B,\mu)\), and let \(w_ t= d(\mu\circ T_ t)/d\mu\). For \(f\in L_ 1(\mu)\), let \(f^*\) denote the maximal function \[ f^*(x)= \sup_{ b>0} \bigl\{\int^ b_ 0 f\circ T_ t(x) w_ t(x) dt\bigl/ \int^ b_ 0 w_ t(x) dt\bigr\}. \] The main result asserts that \[ \alpha\mu(f^*> \alpha)= \int I(f^*> \alpha)f d\mu \] holds for \(\alpha> \int f d\mu\), where \(I\) denotes the indicator function. (This was shown by Marcus and Petersen in the measure preserving case).
Mathematics Subject Classification ID: 28D10
zbMATH DE Number: 633881
zbMATH Keywords: ergodic maximal equality
zbMATH Keywords: nonsingular flows
zbMATH Keywords: measurable flow
zbMATH Keywords: maximal function
author: Ryotaro Sato
reviewed by: Ulrich Krengel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01872096
OpenAlex ID: W1967939785
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description: scientific article; zbMATH DE number 633881