Weighted norm inequalities for geometric fractional maximal operators
type: Item
instance of: scholarly article
title: Weighted norm inequalities for geometric fractional maximal operators
zbMATH Open document ID: 0920.42012
DOI: 10.1007/BF01274188
author: V. Olesen
author: D. Cruz-Uribe
author: C. J. Neugebauer
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-09-09T00:00:00Z
full work available at URL: https://eudml.org/doc/59596
review text: For \(0\leq \alpha<\infty\), let \(T_\alpha f\) denote one of the maximal operators \[ M_{\alpha, 0}f(x)= \sup_{x\in I}| I|^\alpha \exp\Biggl(| I|^{-1} \int_I\log| f|\Biggr), \] \[ M^*_{\alpha, 0}f(x)= \lim_{r\to 0} \sup_{x\in I}| I|^\alpha\Biggl(| I|^{-1} \int_I| f|^r\Biggr)^{1/r}, \] where \(r\to 0\) decreasingly. The authors characterize the pairs of weights \((u,v)\) for which \(T_\alpha\) is a bounded operator from \(L^p(v)\) to \(L^p(u)\), \(0< p\leq q<\infty\). The results extend the norm inequalities for \(\alpha= 0\) in earlier papers [\textit{D. Cruz-Uribe} and \textit{C. J. Neugebauer}, Publ. Math., Barc. 42, No. 1, 239-263 (1998; Zbl 0919.42014)] and [\textit{X. Yin} and \textit{B. Muckenhoupt}, Proc. Am. Math. Soc. 124, No. 1, 75-81 (1986; Zbl 0853.42015)]. As an application, the authors give lower bounds for convolutions \(\Phi*f\), where \(\Phi\) is a rapidly decreasing function.
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1275886
zbMATH Keywords: weighted norm inequalities
zbMATH Keywords: geometric fractional maximal operators
reviewed by: Dashan Fan
MaRDI profile type: Publication
cites work: Weighted Norm Inequalities for the Riemann-Liouville and Weyl Fractional Integral Operators
cites work: A new proof of the two weight norm inequality for the one-sided fractional maximal operator
cites work: The Structure of the Reverse Holder Classes
cites work: Weighted norm inequalities for the geometric maximal operator
cites work: Norm inequalities for the minimal and maximal operator, and differentiation of the integral
cites work: The one-sided minimal operator and the one-sided reverse Holder inequality
cites work: Weighted norm inequalities for a family of one-sided minimal operators
cites work: Q3700300
cites work: Boundedness of Fractional Operators on L p spaces with Different Weights
cites work: Two Weight Norm Inequalities for Fractional One-Sided Maximal Operators
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Weighted Norm Inequalities for Fractional Integrals
cites work: A characterization of a two-weight norm inequality for maximal operators
cites work: Q5640160
cites work: Fractional Integrals on Weighted H p and L p Spaces
cites work: Weighted inequalities for the maximal geometric mean operator
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description: scientific article; zbMATH DE number 1275886