A note on the $L^p$-solvability of a strongly-coupled nonlocal system of equations

The goal of this paper is to study the $L^p$-solvability of the strongly-coupled nonlocal system $$\mathbb{L}\mathbf{u}(\mathbf{x})+\lambda \mathbf{u}(\mathbf{x})=\mathbf{f}(\mathbf{x})\text{in }{\mathbb{R}}^d$$ where $\mathbb{L}$ is a linear nonlocal coupled vector-valued operator associated with a kernel $K$ comparable to $\mathrm{\mid }\mathbf{y}{\mathrm{\mid }}^{-(d+2s)}$ for $s\in (0,1)$, satisfying certain ellipticity and cancellation conditions. For any $\mathbf{f}\in [L^p({\mathbb{R}}^d){]}^d$, $1<p<\mathrm{\infty }$, the existence of a unique strong solution $\mathbf{u}\in [H^{2s,p}({\mathbb{R}}^d){]}^d$ is proved via the method of continuity. To apply this method, we establish the continuity of the operator $\mathbb{L}$ and the necessary $\mathit{\text{a}}\mathit{\text{priori}}$ 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 estimates and a bootstrap argument.