Limits of Dirichlet finite functions along curves
type: Item
instance of: scholarly article
title: Limits of Dirichlet finite functions along curves
zbMATH Open document ID: 0538.31009
DOI: 10.1216/RMJ-1982-12-3-429
author: N. E. Zubov
published in: Rocky Mountain Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: Let R be a noncompact connected real analytic Riemannian n-manifold (perhaps without boundary). Let \(R^*\) be the Royden compactification of R (to which Dirichlet finite functions extend continuously) and let \(\Delta =R^*-R\) be the harmonic part of R. Let \(\tilde M\)(R) be the space of Tonnelli functions on R with finite Dirichlet integrals \(D_ R(f)=\int_{R}df\quad \wedge \quad *\quad df\quad<\quad \infty.\) If H is a family of locally rectifiable curves in R, a non-negative Borel measurable function \(\rho\) is admissible with respect to H if \(\int_{\gamma}\rho ds\geq 1\) for every \(\gamma\in H\) and the modulus of H is given by \(mod H=\inf \{\int_{R}\rho^ 2\quad dV:\quad \rho \quad admissible\}.\) A property is said to hold H everywhere if it holds except on a set \(H_ 0\) of modulus zero. Let B be a fixed ball in R and let G be the family of all curves in R-B joining the boundary of B to the ideal boundary of R. If \(f\in \tilde M(R)\), then \(\lim_{t\to \infty}f(\gamma(t))\) exists for almost all \(\gamma\in G\). The author's main result is then that for \(f\in \tilde M(R)\), \(f(\gamma)=0\) for almost all \(\gamma\in G\) if and only if \(f|_{\Delta}=0\).
reviewed by: Lawrence Gruman
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 3855472
zbMATH Keywords: limits
zbMATH Keywords: noncompact connected real analytic Riemannian n-manifold
zbMATH Keywords: Dirichlet finite functions
zbMATH Keywords: Dirichlet integrals
zbMATH Keywords: ideal boundary
MaRDI profile type: Publication
full work available at URL: rmj-1982-12-3-429
OpenAlex ID: W1976006132
description: scientific article; zbMATH DE number 3855472