On weak type inequalities for dyadic maximal functions
type: Item
instance of: scholarly article
title: On weak type inequalities for dyadic maximal functions
zbMATH Open document ID: 1159.42014
author: Antonios D. Melas
author: Eleftherios N. Nikolidakis
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-10-06T00:00:00Z
review text: Let \(M_df(x)\) be the dyadic maximal function on the euclidean space \(\mathbb R^n\). For \(p>1\) and a cube \(Q\in\mathbb R^n\), let \(\|M_df\|_{L^{p,\infty}(Q)} =\sup_{\lambda>0}\{\lambda \mid \{M_df(x)>\lambda\}\cap Q|^{1/p}\}\) be the local weak-\(L^p\) quasi-norm. The authors consider an extremum problem (or a Bellman function) \(\mathcal B_{p,\infty}(F,f,L)\) \(=\sup\Bigl\{\frac{1}{|Q|}\|M_d\phi\|_{L^{p,\infty}(Q)}^p; \roman{Av}_Q(\phi^p)=F,\, \roman{Av}_Q(\phi)=f,\,\sup_{R;Q\subset R} \roman{Av}_R (\phi)=L\Bigr\}, \) where \(\roman{Av}_Q\,g=\frac{1}{|Q|}\int_Q g(x)\,dx\). They determine the above function and discuss sharp refined weak type estimates for the dyadic maximal functions. To determine the above function, they treat more general families than the family of dyadic cubes, i.e. ``tree'' which has the fundamental properties of dyadic cubes. For these trees, the authors formulate extremum problems and solve them.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 5349100
zbMATH Keywords: maximal functions
zbMATH Keywords: dyadic
zbMATH Keywords: weak type
zbMATH Keywords: Bellman function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.07.001
OpenAlex ID: W2000835098
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DOI: 10.1016/J.JMAA.2008.07.001
description: scientific article; zbMATH DE number 5349100