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 $S$.