An isomorphism theorem for linear evolution problems on the line
type: Item
instance of: scholarly article
title: An isomorphism theorem for linear evolution problems on the line
zbMATH Open document ID: 1055.34109
author: P. J. Rabier
published in: Journal of Dynamics and Differential Equations
publication date: 2004-03-15T00:00:00Z
review text: The author considers the problem \[ {du\over dt}- Au= f,\tag{1} \] where \(X\) is a complex Banach space, \(u: \mathbb{R}\to X\), \(A\) is a closed linear operator on \(X\) with dense domain and \(f\in L^p(\mathbb{R}, X)\) for \(p\in (1,\infty)\). Weis' \(L^p\) multiplier theorem [\textit{L. Weis}, Math. Ann. 319, 735--758 (2001; Zbl 0989.47025)] is used to prove existence and uniqueness of solutions of (1) under the assumptions that \(X\) has ``unconditionality of martingale differences'', the spectrum of \(A\) contains no purely imaginary numbers, plus an assumption on the resolvent of \(A\) which involves the Rademacher bound. Rademacher boundedness is a key concept for the Weis multiplier theorem. Applications are discussed in section 5, including Hamiltonian systems. The concepts mentioned above which are probably not familiar to all readers are discussed in detail. The author makes very detailed connections with the literature and includes 62 references.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 2057849
zbMATH Keywords: multiplier theorem
zbMATH Keywords: \(r\)-boundedness
zbMATH Keywords: UMD space
zbMATH Keywords: Bochner integral
zbMATH Keywords: Hamiltonian system
MaRDI profile type: Publication
full work available at URL: b:jody.0000010065.11894.b6
OpenAlex ID: W1988993685
DOI: 10.1023/B:JODY.0000010065.11894.B6
description: scientific article; zbMATH DE number 2057849