Arendt, Wolfgang; Bu, Shangquan The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. (English) Zbl 1018.47008 Math. Z. 240, No. 2, 311-343 (2002). The non-homogeneous problem \(u'+Au=f\) with periodic boundary conditions and a closed linear operator on a UMD-space is considered. The authors characterize the maximal \(L^p\) regularity of the problem in terms of R-boundedness of the resolvent. This approach was applied by L. Weis [Lect. Notes Pure Appl. Math. 215, 195-214 (2001; Zbl 0981.35030)] in the case of Dirichlet boundary conditions when \(A\) is an infinitesimal generator of a bounded analytic semigroup on a UMD-space. The present paper is a generalization of these results. The main tool (with respect to the periodic boundary conditions) is a discrete analog of the Marcinkiewicz operator-valued multiplier theorem. The authors present a direct and easy proof of this theorem. One of the main results of the paper is that the considered problem is strongly \(L^p\)-well-posed for \(1<p<\infty\) if and only if the set \(\{k(ik-A)^{-1}:k\in\mathbb{Z}\}\) is R-bounded. The maximal regularity of the second order problem for periodic, Dirichlet or Neumann boundary conditions is also characterized. Reviewer: Teresa Regińska (Warszawa) Cited in 6 ReviewsCited in 106 Documents MSC: 47A50 Equations and inequalities involving linear operators, with vector unknowns 35K90 Abstract parabolic equations 34G10 Linear differential equations in abstract spaces Keywords:\(R\)-boundedness; discrete multiplier theorem; maximal regularity; strong \(L^p\)-well-posedness; mild solution Citations:Zbl 0981.35030 PDFBibTeX XMLCite \textit{W. Arendt} and \textit{S. Bu}, Math. Z. 240, No. 2, 311--343 (2002; Zbl 1018.47008) Full Text: DOI