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.
Abbate et al. (Fri,) studied this question.