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.
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© 2004 Springer-Verlag Berlin/Heidelberg
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Kunstmann, P.C., Weis, L. (2004). Maximal L p -regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\)-functional Calculus. In: Iannelli, M., Nagel, R., Piazzera, S. (eds) Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol 1855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44653-8_2
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DOI: https://doi.org/10.1007/978-3-540-44653-8_2
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Publisher Name: Springer, Berlin, Heidelberg
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