Piecewise monotonic doubling measures
type: Item
instance of: scholarly article
title: Piecewise monotonic doubling measures
zbMATH Open document ID: 0861.42007
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-04-23T00:00:00Z
full work available at URL: CONT26-2.html
review text: The author studies weights (\(A_p\) weights, reverse Hölder weights, and doubling weights) under the restriction of piecewise monotonicity of the density. The \(A_p\) weights control mapping properties of the Hardy-Littlewood maximal operator and of the Hilbert transform. The spaces are defined by \(w\in A_p\) if and only if \[ A^p_p(w)= \sup_Q \Biggl({1\over {|Q|}} \int_Q w\Biggr) \Biggl({1\over {|Q|}} \int_Q w^{-1/(p-1)} \Biggr)^{p-1}< +\infty. \] A measure \(\mu\) is doubling if there is a constant \(C\) such that for each interval \(I\), \(\mu(2I)\leq C\mu(I)\), where \(2I\) is the interval with the same center as \(I\) and twice the length. Doubling measures are not necessarily absolutely continuous. \(A_\infty\) weights are doubling, but not every doubling weight is an \(A_\infty\) weight. The \(A_p\) weights form a nested set with \[ A_1\subseteq A_p\subseteq A_q\subseteq A_\infty= \bigcup_{1\leq p\leq\infty} A_p\subseteq D, \qquad 1\leq p\leq q\leq\infty. \] The author shows that piecewise monotonic doubling measures are absolutely continuous, which means that results can be expressed in terms of the density of the measure. He first characterizes decreasing and increasing doubling weights. Several characterizations are given; I will cite ones in terms of the averaging operator \(A\), where \(Aw(t)= {1\over t}\int^t_0 w(x)dx\). A decreasing weight \(w\) is doubling if and only if there is a constant \(C\) such that \(Aw(t)\leq Cw(t)\), for all \(t>0\) if and only if \(w\in A_1\). An increasing weight \(w\) is doubling if and only if there is a constant \(C\) such that \(CAw(t)\geq w(t)\), for all \(t>0\) if and only if \(w\in A_\infty\). Using these results among others, he reproves results of Guseinov and of Benedetto, Heinig and Johnson characterizing monotone \(A_p\) weights. He also gives necessary and sufficient conditions for monotonic functions to be pointwise multipliers of the class of monotonic weights. He proves that \(\phi\) is a multiplier of dec…
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 944407
zbMATH Keywords: reverse Hölder weights
zbMATH Keywords: doubling weights
zbMATH Keywords: \(A_ p\) weights
zbMATH Keywords: Hardy-Littlewood maximal operator
zbMATH Keywords: Hilbert transform
zbMATH Keywords: piecewise monotonic doubling measures
zbMATH Keywords: density
zbMATH Keywords: multipliers
author: D. Cruz-Uribe
reviewed by: Raymond Johnson
MaRDI profile type: Publication
OpenAlex ID: W2018813793
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DOI: 10.1216/RMJM/1181072073
description: scientific article; zbMATH DE number 944407