Minimally fine limits at infinity for \(p\)-precise functions
type: Item
instance of: scholarly article
title: Minimally fine limits at infinity for \(p\)-precise functions
zbMATH Open document ID: 1052.31003
author: Yoshihiro Mizuta
author: Tetsu Shimomura
published in: Journal of the Mathematical Society of Japan
publication date: 2004-08-16T00:00:00Z
review text: If \(u\geq 0\) is a superharmonic function in the half-plane \(D= \mathbb{R}^{n-1}\times \{x_n>0\}\), then \(u(x)=ax_n+G\mu (x)+P\nu (x)\), where the constant \(a\geq 0\) and \(G\mu\) and \(P\nu\) are, respectively, Green and Poisson potentials of nonnegative measures \(\mu\) and \(\nu\). This representation implies that \(\lim u(x)/x_n=a\) when \(D\ni x\rightarrow \infty\) outside an exceptional set that is minimally thin at infinity in \(D\). Considering more refined characteristics of an exceptional set, one can make this statement more precise and, in particular, estimate the rate of convergence of \(u(x)/x_n\) to \(a\) [see, e.g., \textit{H. Aikawa} and \textit{M. Essén}, Potential theory -- selected topics (1996; Zbl 0865.31001)]. The present authors extend these results to locally \(p\)-precise functions in \(D\) satisfying \(\int _D | \nabla u| ^p x_n^{\gamma}\,dx<\infty\), \(0<p<\infty\), \(-1<\gamma <p-1\). Here a function \(u\in W^1_p(\Omega)\) is called \(p\)-precise in a bounded domain \(\Omega\) [\textit{W. Ziemer}, Mich. Math. J. 17, 117--128 (1970; Zbl 0183.39104)] if for each \(\varepsilon >0\), there exists an open set \(U\) such that \(u| _{\Omega \setminus U}\) is continuous and \(\psi _p(U)<\varepsilon\), where \((\psi _p(U))^{-1}\) is the \(p\)-dimensional extremal length of all closed connected sets joining \(U\) to infinity at \(\mathbb{R}^n\).
reviewed by: Alexander I. Kheyfits
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 31C40
zbMATH DE Number: 2092004
zbMATH Keywords: locally \(p\)-precise functions
zbMATH Keywords: fine limits
zbMATH Keywords: superharmonic function
zbMATH Keywords: extremal length
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191418692
OpenAlex ID: W2169796672
DOI: 10.2969/JMSJ/1191418692
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description: scientific article; zbMATH DE number 2092004