The behavior of the bounds of matrix-valued maximal inequality in \(\mathbb{R}^{n}\) for large \(n\)
type: Item
instance of: scholarly article
title: The behavior of the bounds of matrix-valued maximal inequality in \(\mathbb{R}^{n}\) for large \(n\)
zbMATH Open document ID: 1311.42050
author: Guixiang Hong
published in: Illinois Journal of Mathematics
publication date: 2014-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.1245
full work available at URL: https://projecteuclid.org/euclid.ijm/1415023514
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46L51
zbMATH DE Number: 6369194
zbMATH Keywords: matrix-valued maximal functions
zbMATH Keywords: weak type \((1, 1)\) bound
zbMATH Keywords: vector-valued noncommutative \(L^p\)-spaces
MaRDI profile type: Publication
arXiv ID: 1411.1245
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review text: Let \((X,d,\mu)\) be a metric measure space and \(\mathcal B(\ell^2)\) the matrix algebra of bounded operators on \(\ell^2\). For a locally integrable \(\mathcal B(\ell^2)\)-valued function \(f\), define \(f_r(x) =\mu(B(x, r))^{-1}\int_{B(x,r)}f(y)d\mu(y)\), where \(B(x, r) = \{y\in X : d(x, y)\leq r\}\).NEWLINENEWLINEIn this paper, the author studies the behavior of the bounds of matrix-valued maximal functions in \(\mathbb R^n\) for large \(n\). The main result is that the \(L^p\)-bounds \((p > 1)\) can be taken to be independent of \(n\), which is a generalization of Stein and Strömberg's result in the scalar-valued case. It is also shown that the weak type \((1, 1)\) bound has similar behavior as Stein and Strömberg's one.NEWLINENEWLINEHe points out that Stein and Strömberg's result in the scalar-valued case seems not to be available for matrix-valued functions since one cannot compare any two matrices or operators, which is a source of difficulties in the noncommutative analysis. But the author states several histories how the obstacle has been successfully overcome by the interaction with operator space theory, to go from the scalar-valued case to the matrix-valued or more generally to the operator-valued case.
description: scientific article; zbMATH DE number 6369194