Jensen measures and maximal functions of uniform algebras
type: Item
instance of: scholarly article
title: Jensen measures and maximal functions of uniform algebras
zbMATH Open document ID: 0609.46029
author: Cho-ichiro Matsuoka
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: For any uniform algebra A and any point q of the maximal ideal space of A the author seeks a certain class \(T_ q\) of Jensen measures for q which admit a maximal function to any long-envelope function associated with A. The maximal functions can be considered as a proper generalization of Brownian maximal functions well-known in the classical Hardy spaces. Most of the results already obtained from the probabilistic approach, e.g. Burkholder-Gundy-Silverstein inequalities, Fefferman-Stein estimate, etc., are valid for our maximal functions. The remarkable feature of \(T_ q\) is that it is stable in the convex combination and weakstar limit. Various examples are discussed in the concrete algebra R(K). Further results are known: for any A and any q the class \(T_ q\) contains a measure which is supported on the Shilov boundary of A. Moreover, in the case \(A=R(K)\) the totality of Jensen measures for q is identical with \(T_ q\). The definition of \(T_ q\) is applicable under more general situation, e.g. any compact convex set in some locally convex topological vector space and the cone of continuous convex functions on that set. Then every probability measure on the convex set belongs to the class similar to \(T_ q\), where q is equal to the barycenter of the measure in question.
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 60G46
zbMATH DE Number: 3985930
zbMATH Keywords: uniform algebra
zbMATH Keywords: maximal ideal space
zbMATH Keywords: Jensen measures
zbMATH Keywords: long-envelope function
zbMATH Keywords: Brownian maximal functions
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Shilov boundary
zbMATH Keywords: cone of continuous convex functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178372
OpenAlex ID: W2055383175
cites work: Q5611281
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DOI: 10.2977/PRIMS/1195178372
description: scientific article; zbMATH DE number 3985930