Harmonic calculus on fractals -- a measure geometric approach. I
type: Item
instance of: scholarly article
title: Harmonic calculus on fractals -- a measure geometric approach. I
zbMATH Open document ID: 1055.28002
DOI: 10.1023/A:1014085203265
published in: Potential Analysis
publication date: 2002-05-21T00:00:00Z
review text: Differentiation, \(\frac{df}{d\mu}\), of functions \(f\) w.r.t. finite atomless measures \(\mu\) with compact support on the real line is considered. The derivative is defined via the fundamental theorem of calculus, where the Lebesgue measure is replaced by \(\mu\). The ``Laplacian'' then is \(\frac{d^2f}{d\mu^2}\). Its spectral asymptotics in \(L^2([0,1], \mu)\) has Weyl's classical form when the measure is supported by a Cantor set with arbitrary weights. The main technical point is a scaling property of the resolvent operator.
reviewed by: Volker Metz
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 1742919
zbMATH Keywords: measure derivative
zbMATH Keywords: Weyl asymptotics
zbMATH Keywords: fractals
zbMATH Keywords: Cantor set
author: Uta Freiberg
author: Martina Zähle
MaRDI profile type: Publication
full work available at URL: a:1014085203265
OpenAlex ID: W1637352950
description: scientific article; zbMATH DE number 1742919