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The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. (English) Zbl 1018.47008
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 {\it 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 $\{k(ik-A)^{-1}:k\in\bbfZ\}$ is R-bounded. The maximal regularity of the second order problem for periodic, Dirichlet or Neumann boundary conditions is also characterized.

47A50Equations and inequalities involving linear operators, with vector unknowns
35K90Abstract parabolic equations
34G10Linear ODE in abstract spaces
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