an:02144723
Zbl 1097.47041
Kunstmann, Peer C.; Weis, Lutz
Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty\)-functional calculus
EN
Iannelli, Mimmo (ed.) et al., Functional analytic methods for evolution equations. Based on lectures given at the autumn school on evolution equations and semigroups, Levico Terme, Trento, Italy, October 28--November 2, 2001. Berlin: Springer (ISBN 3-540-23030-0/pbk). Lecture Notes in Mathematics 1855, 65-311 (2004).
2004
a
47D06 47A60 34G10 35D10 35J55 35K20 35K90 42B20
parabolic equations; maximal regularity
The paper under review is a set of lecture notes on recent progress in the functional analytic approach to maximal regularity for parabolic evolution equations with extensive applications to maximal \(L_p\)-regularity of large classes of partial differential operators and systems. The authors describe two approaches to maximal regularity: singular integrals and \(H^\infty\)-calculus. They provide effective Mihlin multiplier theorems in UMD-spaces, and as a consequence, characterize maximal regularity in terms of \(R\)-bounededness. Then the theory is applied to classical operators, elliptic systems, boundary value problems, and divergence type elliptic operators. In the second part, the authors construct an \(H^\infty\)-calculus of sectorial operators, characterize its boundedness, provide connections with the ``operator sum'' method and with \(R\)-boundedness. They prove the boundedness of the \(H^\infty\)-calculus for various classes of differential operators. An appendix provides the necessary background on fractional powers of sectorial operators.
For the entire collection see [Zbl 1052.47002].
V. A. Liskevich (Bristol)