A note on the Lp-solvability of a strongly coupled nonlocal system of equations

The goal of this paper is to study the [Formula: see text]-solvability of the strongly coupled nonlocal system [Formula: see text] where [Formula: see text] is a linear nonlocal coupled vector-valued operator associated with a kernel [Formula: see text] comparable to [Formula: see text] for [Formula: see text], satisfying certain ellipticity and cancellation conditions. For any [Formula: see text], [Formula: see text], the existence of a unique strong solution [Formula: see text] is proved via the method of continuity. To apply this method, we establish the continuity of the operator [Formula: see text] and the necessary a priori estimates. These are obtained through the study of the corresponding parabolic system. The proof strategy follows and extends recent ideas developed for the scalar setting, combining commutator estimates, Sobolev embeddings, a level set estimate and a bootstrap argument.