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Maximal $$L_p$$-regularity for parabolic equations, Fourier multiplier theorems and $$H^\infty$$-functional calculus. (English) Zbl 1097.47041
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).
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].

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47A60 Functional calculus for linear operators 34G10 Linear differential equations in abstract spaces 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35K20 Initial-boundary value problems for second-order parabolic equations 35K90 Abstract parabolic equations 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
##### Keywords:
parabolic equations; maximal regularity