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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.

##### MSC:
 47A50 Equations and inequalities involving linear operators, with vector unknowns 35K90 Abstract parabolic equations 34G10 Linear ODE in abstract spaces
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