Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-04T21:58:46.678Z Has data issue: false hasContentIssue false

Tools for maximal regularity

Published online by Cambridge University Press:  01 May 2003

WOLFGANG ARENDT
Affiliation:
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany. e-mail: [email protected]
SHANGQUAN BU
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, P. R. of China. e-mail: [email protected]

Abstract

Let A be the generator of an analytic C0-semigroup on a Banach space X. We associate a closed operator ${\cal A}_{1}$ with A defined on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator ${\cal A}_{1}{\rm g}$ generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that ${\cal R}$-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is ${\cal R}$-bounded if and only if it is ${\cal R}$-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every $f\in L^{p}(\RR_{+}; X)$, the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs to $W^{\theta,p}(\RR_{+}; X)$ for every $0<\theta < 1$.

Type
Research Article
Copyright
2003 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research is part of the DFG-project: ‘Regularität und Asymptotik für elliptische und parabolische Probleme’. The second author is supported by the Alexander-von-Humboldt Foundation and the NSF of China.