Weyl type bounds for harmonic functions
type: Item
instance of: scholarly article
title: Weyl type bounds for harmonic functions
zbMATH Open document ID: 0919.31004
published in: Inventiones Mathematicae
publication date: 1998-06-11T00:00:00Z
review text: Let \((M^n,g)\) be a complete Riemannian manifold having the doubling property and satisfying the scale invariant Neumann-Poincaré inequality. Let \({\mathcal H}_d\), \(d\geq 1\), be the linear space of harmonic functions \(u\) on the open manifold \(m^n\) (which can be defined by the Laplacian or more generally by means of a quasi-uniformly elliptic operator) such that \(| u|\leq c(1+r^d)\) holds, where \(r\) is the distance from a fixed point. There is a nice estimate here for the dimension of \({\mathcal H}_d\). Moreover, similar to the Weyl formula for eigenfunctions on compact domains in the Euclidean space, in this interesting article the authors present an asymptotic formula and a uniform estimate on the remainder for harmonic functions with polynomial growth on open manifolds with nonnegative Ricci curvature.
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 1145860
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: space of harmonic functions
zbMATH Keywords: dimension
zbMATH Keywords: asymptotic formula
zbMATH Keywords: uniform estimate
zbMATH Keywords: remainder
zbMATH Keywords: harmonic functions with polynomial growth
zbMATH Keywords: nonnegative Ricci curvature
author: Tobias Holck Colding
author: William P. II Minicozzi
reviewed by: Victor Anandam
MaRDI profile type: Publication
full work available at URL: s002220050204
OpenAlex ID: W2028721699
DOI: 10.1007/S002220050204
description: scientific article; zbMATH DE number 1145860