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Maximal Lp-regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\)-functional Calculus

  • Peer C. KunstmannEmail author
  • Lutz Weis
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1855)

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 2-8) we concentrate on the singular integral approach to maximal regularity. In particular we present effective Mihlin multiplier theorems for operator-valued multiplier functions in UMD-spaces as an interesting blend of ideas from the geometry of Banach spaces and harmonic analysis with R-boundedness at its center. As a corollary of this result we obtain a characterization of maximal regularity in terms of R-boundedness. We also show how the multiplier theorems “bootstrap” to give the R-boundedness 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 9-15) we construct the \(H^\infty\)-calculus, give various characterizations for its boundedness, and explain its connection with the “operator-sum” method and R-boundedness. 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 

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Mathematisches Institut IUniversität KarlsruheKarlsruheGermany

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