The following question is motivated from Chapter 2 (Generalized Hodge Systems in 2D), particularly Section 2.3 ($L^p$ theory for Hodge systems in 2D) of Christodoulou and Klainerman's book, *The global nonlinear stability of the Minkowski space.*

In this section (page 43, in my copy), the authors state that the Calderon-Zygmund inequalities on the standard unit round 2-sphere imply that for a Hodge system of the form 
$$
\text{div}\xi = f \\
\text{curl}\xi = g
$$
one has the estimates
$$
\int_S |\nabla \xi|^p + |\xi|^p \leq C_p\int_S |f|^p + |g|^p.
$$
Here, $S = S^2$ is the standard unit round sphere, and $\xi$ is a 1-form on $S$, while $f, g$ are scalar functions on $S$. 

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.