Harmonic majorization and Brownian motion
type: Item
instance of: scholarly article
title: Harmonic majorization and Brownian motion
zbMATH Open document ID: 0791.31001
author: Jean Brossard
author: Lucien Chevalier
published in: Illinois Journal of Mathematics
publication date: 1994-05-16T00:00:00Z
review text: Using probabilistic methods, we first give a very simple proof of the following recent theorem of \textit{M. Essén} [Ann. Acad. Sci. Fenn., Ser. A I 15, No. 1, 83-105 (1990; Zbl 0696.31008)]: Let \(D\) be a domain in \(\mathbb{R}^ n\), \(n \geq 2\). If \(p > 1\), then the subharmonic function \(x \to | x|^ p\) has a harmonic majorant in \(D\) if and only if the function \(x_ 1 \to | x_ 1|^ p\) has one, where \(x_ i\) denotes the \(i\)-th coordinate of \(x\). If \(p \leq 1\), the above equivalence is no longer true; in this case, we obtain a similar and new result, where \(| x_ 1|^ p\) is replaced by \((| x_ 1|^ 2 + | x_ 2|^ 2)^{p/2}\). Moreover, we answer some open question relative to the special case \(p = 1\), and obtain a new necessary and sufficient condition for \(x \to | x_ 1| \log^ + | x_ 1 |\) to have a harmonic majorant in \(D\). Our proofs involve classical relations between potential theory and Brownian motion, Burkholder-Gundy inequalities for martingales, Burkholder's results on harmonic majorization [\textit{D. L. Burkholder}, Adv. Math. 26, 182-205 (1977; Zbl 0372.60112)] and the authors' recent characterization of the class \(L \log L\) [\textit{J. Brossard} and \textit{L. Chevalier}, C. R. Acad. Sci., Paris, Ser. I 305, 135-137 (1987; Zbl 0624.60057)].
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 561189
zbMATH Keywords: harmonic majorization
zbMATH Keywords: \(H^ p\) spaces
zbMATH Keywords: class \(L\) Log \(L\)
zbMATH Keywords: Burkholder- Gundy inequalities for martingales
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 561189