The operator-valued Marcinkiewicz multiplier theorem and maximal regularity
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 \textit{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.
scientific article; zbMATH DE number 1801561