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 $S$. The divergence and curl are 
$$
\text{div} \xi = g^{AB}\nabla_A \xi_B, \ \ \ \ \text{curl}\xi = \epsilon^{AB}\nabla_A \xi_B
$$
where $g$ is the unit round metric on $S$ and $\epsilon$ is the corresponding volume form.

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.

For a more complete summary, here is the statement of the lemma.

**Lemma 2.3.1.** Let $\xi$ be a 1-form on $S = S^2$ solving the equations
$$
\text{div} \xi = f 
$$
$$
\text{curl} \xi = g.
$$
For every $1 < p < \infty$, there exists a constant $C_p$ such that
$$
\int_S |\nabla \xi|^p + |\xi|^p \leq C_p\int_S |f|^p + |g|^p
$$
and
$$
\int_S |\nabla^2 \xi|^p \leq C_p\int_S |\nabla f|^p + |\nabla g|^p + |f|^p + |g|^p.
$$
All integrals are with respect to the volume form on $S$.