Functional Analytic Methods for Evolution Equations
Volume 1855 of the series Lecture Notes in Mathematics pp 65311
Maximal L_{ p }regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\)functional Calculus
 Peer C. KunstmannAffiliated withMathematisches Institut I, Universität Karlsruhe Email author
 , Lutz WeisAffiliated withMathematisches Institut I, Universität Karlsruhe
Abstract
In these lecture notes we report on recent breakthroughs in the functional analytic approach to maximal regularity for parabolic evolution equations, which set off a wave of activity in the last years and allowed to establish maximal L_{ p }regularity for large classes of classical partial differential operators and systems.
In the first chapter (Sections 28) we concentrate on the singular integral approach to maximal regularity. In particular we present effective Mihlin multiplier theorems for operatorvalued multiplier functions in UMDspaces as an interesting blend of ideas from the geometry of Banach spaces and harmonic analysis with Rboundedness at its center. As a corollary of this result we obtain a characterization of maximal regularity in terms of Rboundedness. We also show how the multiplier theorems “bootstrap” to give the Rboundedness of large classes of classical operators. Then we apply the theory to systems of elliptic differential operators on \(\mathbb{R}^n\) or with some common boundary conditions and to elliptic operators in divergence form.
In Chapter II (Sections 915) we construct the \(H^\infty\)calculus, give various characterizations for its boundedness, and explain its connection with the “operatorsum” method and Rboundedness. In particular, we extend McIntosh’s square function method form the Hilbert space to the Banach space setting. With this tool we prove, e.g., a theorem on the closedness of sums of operators which is general enough to yield the characterization theorem of maximal L_{ p }regularity. We also prove perturbation theorems that allow us to show boundedness of the \(H^\infty\)calculus for various classes of differential operators we studied before. In an appendix we provide the necessary background on fractional powers of sectorial operators.
Mathematics Subject Classification (2000):
34Gxx 34K30 35K90 42A45 47Axx 47D06 47D07 49J20 60J25 93B28 Title
 Maximal L_{ p }regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\) functional Calculus
 Book Title
 Functional Analytic Methods for Evolution Equations
 Pages
 pp 65311
 Copyright
 2004
 DOI
 10.1007/9783540446538_2
 Print ISBN
 9783540230304
 Online ISBN
 9783540446538
 Series Title
 Lecture Notes in Mathematics
 Series Volume
 1855
 Series ISSN
 00758434
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin/Heidelberg
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 Peer C. Kunstmann ^{(1)}
 Lutz Weis ^{(1)}
 Author Affiliations

 1. Mathematisches Institut I, Universität Karlsruhe, Englerstrasse 2, D76128, Karlsruhe, Germany
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