Maximal averages over linear and monomial polyhedra
type: Item
instance of: scholarly article
title: Maximal averages over linear and monomial polyhedra
zbMATH Open document ID: 1197.42011
DOI: 10.1215/00127094-2009-039
author: Alexander Nagel
author: Malabika Pramanik
published in: Duke Mathematical Journal
publication date: 2009-08-25T00:00:00Z
review text: Let \(\mathfrak P=\{\mathbf p_1,\dots,\mathbf p_d\}\) be a spanning set of vectors in \(\mathbb R^n\), let \(F_p(\mathbf x)=x_1^{p_1}\dots x_n^{p_n}\), where \(\mathbf p=(p_1,\dots,p_n)\), and consider the set of monomials \(\mathcal P=\{F_{\mathbf p_1}, \dots ,F_{\mathbf p_d}\}\). Let \(\mathcal O_n=\{\mathbf x=(x_1,\dots ,x_n)\in \mathbb R^n\mid x_j >0,1\leq j \leq n\}\). For every \(\mathbf x\in \mathcal O_n\) and every \(\overline \delta =(\delta_1,\dots ,\delta_d)\) with \(\delta_j>0, 1\leq j \leq d\), define \(B_{\mathcal P}(\mathbf x;\overline \delta)\) be the closure of the connected component of the interior of the set \(\{\mathbf y\in \mathcal O_n \mid |F_{p_j}(\mathbf y)-F_{pj}(\mathbf x)|\leq \delta_j, 1\leq j\leq d\}\) which contains \(\mathbf x\). Define the maximal operator \[ \mathcal M_{\mathcal P}[f](\mathbf x)=\sup_{\overline \delta}|B_{\mathcal P}(\mathbf x;\overline \delta)|^{-1}\int_{B_{\mathcal P}(\mathbf x;\overline \delta)}|f(\mathbf y)|d\mathbf y. \] The goal of this paper is to establish the \(L^p\) boundedness of this maximal operator. In fact, the authors also consider the \(L^p\) boundedness of the following more general maximal operators: Let \[ \mathbb W_{\mathfrak P}(\overline \alpha, \overline \beta)=\{\mathbf x \in \mathcal O_n \mid \alpha_j \leq F_{\mathbf p}(x) \leq \beta_j,1 \leq j \leq d\} \] and define \[ \mathcal Mf(\mathbf x) = \sup _{\mathbb W_{\mathfrak P}}\mu_{\mathbf e}(\mathbb W_{\mathfrak P})^{-1}\int_{\mathbb W_{\mathfrak P}}|f(\mathbf {xy})|d\mu_{\mathbf e}(\mathbf y). \] The idea is to construct a superset \(\mathcal N\) of \(\mathbb W\) such that \(\mathcal N\) and \(\mathbb W\) have comparable measures, and the maximal operator defined by the \(\mathcal N\)'s are comparable with the classical strong maximal operators. The construction requires a detailed study of the geometric properties of these monomial polyhedra. With some restriction on the monomial polyhedra, the authors establish the following main results o…
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 32A40
zbMATH DE Number: 5598217
zbMATH Keywords: monomial polyhedra
zbMATH Keywords: maximal operator
zbMATH Keywords: \(L^p\)-boundedness
reviewed by: Lee Taichung
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2009-039
OpenAlex ID: W1978359906
cites work: Q4790110
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description: scientific article; zbMATH DE number 5598217