In the global nonlinear stability of Minkowski space (page 43, in my copy), the authors state that the Calderon-Zygmund inequalities on the sphere imply that for a Hodge system of the form $$ \text{div}\xi = f \\ \text{curl}\xi = g $$ one has the estimates $$ \int_S |\nabla \xi|^p + |\xi|^p \leq C_p\int_S |f|^p + |g|^p. $$ I'm looking for a reference for this fact and the theory behind it. I have only seen Calderon-Zygmund inequalities briefly in the context of Euclidean space $\mathbb{R}^d$, and I have not seen them applied to Hodge systems.