On Kolmogorov's theorem, the Hardy-Littlewood maximal function and the radial maximal function
type: Item
instance of: scholarly article
title: On Kolmogorov's theorem, the Hardy-Littlewood maximal function and the radial maximal function
zbMATH Open document ID: 0658.42020
DOI: 10.1007/BF02796124
author: Joaquim Bruna
author: Boris Korenblum
published in: Journal d'Analyse Mathématique
publication date: 1988-01-01T00:00:00Z
review text: The main object of study in this paper is the weak maximal function. This is the function \(M_ w\phi\) defined for any weak integrable function \(\phi\) as \(M_ w\phi (x)=\sup \{\| \phi \|^ I_{l,\infty}/| I|:x\in I\}\), where \(\| \phi \|^ I_{l,\infty}\) denotes the \(``weak\)-L\({}^ 1\) norm'' of \(\phi\) on the interval I, i.e., \(\| \phi \|^ I_{l,\infty}=\sup \{\delta | \{t\in I:| \phi (t)| >\delta \}|;\delta >0\}\). In case \(\phi\) is the modulus of the boundary values of an holomorphic function f with positive real part, it is shown that \(M_ w\phi\) is pointwise equivalent to the radial maximal function of f. The main result of the paper is that the weak (1,1) estimate for both the Hardy-Littlewood maximal function and the conjugate function of a measure \(\mu\) can be improved in the following sense: if \(\phi\) is one of those functions, then \(\| M^ m_ w\phi \|_{l,\infty}\leq C_ 1C_ 2^ m\| \mu \|\), \(m=1,2,...\). for some absolute constants \(C_ 1,C_ 2\). This property does not hold for an arbitrary weak-integrable function. The generalization of this result to a Calderón-Zygmund singular convolution operator appeared in Bull Am. Math. Soc., New Ser. 16, 271-273 (1987; Zbl 0617.42010).
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 4075749
zbMATH Keywords: weak maximal function
zbMATH Keywords: radial maximal function
zbMATH Keywords: Hardy-Littlewood maximal function
zbMATH Keywords: conjugate function
zbMATH Keywords: Calderón-Zygmund singular convolution operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4075749