The non-homogeneous problem
with periodic boundary conditions and a closed linear operator on a UMD-space is considered. The authors characterize the maximal
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
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
if and only if the set
is R-bounded. The maximal regularity of the second order problem for periodic, Dirichlet or Neumann boundary conditions is also characterized.