Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators
type: Item
instance of: scholarly article
title: Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators
zbMATH Open document ID: 1362.30052
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-02-08T00:00:00Z
review text: The theory of the homogeneous spaces \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) on \(\mathbb R^n\) with \(s\in\mathbb R\), \(0<p,q \leq \infty\) (\(p<\infty\) for \(F\)-spaces) can be developed on the basis of the (essentially) self-adjoint Laplacian \(L = -\Delta\) in \(L_2 (\mathbb R^n)\), its powers \((-\Delta)^m\) and, in particular, the related Gauss-Weierstrass semi-group \(e^{-tL} = e^{t\Delta}\), \(t>0\), and their kernels with the well-known miraculous properties. Spectral theory, that is \(f(L)\) for suitable real bounded measurable functions \(f\) on \(\mathbb R_+\) (preferably cut-off functions) can be taken as a starting point to construct suitable frame representations for \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) on \(\mathbb R^n\) in the context of the dual pairing \(\big( \dot{S} (\mathbb R^n), \dot{S}' (\mathbb R^n) \big)\) (modulo polynomials). The paper extends this approach to metric measure spaces \((M,\rho, \mu)\) where \(\rho\) is a distance and \(\mu\) a doubling Radon measure. It is assumed that there is a substitute \(L\) of the above Laplacian, a self-adjoint non-negative operator in \(L_2 (M, d\mu)\), imitating crucial properties of the Gauss-Weierstrass semi-group and the related kernels. On this basis the authors develop a theory of corresponding homogeneous spaces \(\dot{B}^s_{p,q} (L)\) and \(\dot{F}^s_{p,q} (L)\) (in two versions), including frame representations.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 30L99
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6682102
zbMATH Keywords: metric measure spaces
zbMATH Keywords: doubling Radon measure
zbMATH Keywords: Laplacian
zbMATH Keywords: Besov spaces
zbMATH Keywords: Triebel-Lizorkin spaces
zbMATH Keywords: homogeneous spaces
author: Athanasios G. Georgiadis
author: Gérard Kerkyacharian
author: G. Kyriazis
author: Pencho Petrushev
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.12.049
OpenAlex ID: W2563239557
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DOI: 10.1016/J.JMAA.2016.12.049
description: scientific article; zbMATH DE number 6682102