The maximal regularity property of abstract integro-differential equations

We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces $\Phi$ and/or the scales of Besov and Triebel-Lizorkin spaces defined by $\Phi$. We apply this result to the study of the well-posedness and maximal regularity property of abstract second-order integro-differential equation, which models various types of elliptic and parabolic problems arising in different areas of applied mathematics.