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].