The operator-valued Marcinkiewicz multiplier theorem and maximal regularity

Abstract.

Given a closed linear operator on a UMD-space, we characterize maximal regularity of the non-homogeneous problem

$u' + Au = f$

with periodic boundary conditions in terms of R-boundedness of the resolvent. Here A is not necessarily generator of a $C_0$-semigroup. As main tool we prove an operator-valued discrete multiplier theorem. We also characterize maximal regularity of the second order problem for periodic, Dirichlet and Neumann boundary conditions.