  By methods of harmonic analysis, we identify large classes of Banach spaces
invariant of periodic Fourier multipliers with symbols satisfying the classical
Marcinkiewicz type conditions. Such classes include general (vector-valued)
Banach function spaces $\Phi$ and/or the scales of Besov and Triebel-Lizorkin
spaces defined on the basis of $\Phi$.
  We apply these results to the study of the well-posedness and maximal
regularity property of an abstract second-order integro-differential equation,
which models various types of elliptic and parabolic problems arising in
different areas of applied mathematics. In particular, under suitable
conditions imposed on a convolutor $c$ and the geometry of an underlying Banach
space $X$, we characterize the conditions on the operators $A$, $B$ and $P$ on
$X$ such that the following periodic problem ${\partial P} {\partial u} + B
{\partial u} + A u + c \ast u = f \qquad \textrm{in } \mathcal{D}'(\mathbb{T};
X)$ is well-posed with respect to large classes of function spaces. The
obtained results extend the known theory on the maximal regularity of such
problem.
