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 $S$.



Reference for Calderon-Zygmund $L^p$ inequalities on the sphere
I don't know of an exact reference, and in general this sort of result (transfering a "classical" result from the analysis of PDE in $\mathbb{R}^n$ to Riemannian manifolds) is often quite hard to track down. Instead, I can explain how you derive the inequalities you want from the standard Calderon-Zygmund estimates for domains in $\mathbb{R}^n$. What follows is quite long for a MathOverflow answer, but I've tried to be as complete as possible. It is a "standard" patching argument to turn estimates from $\mathbb{R}^n$ into estimates on a manifold. I don't remember where I learned this sort of argument from (maybe Wells' book on analysis on complex manifolds?) but it is ubiquitous in geometric analysis.


Take a deeep breath, and here we go.


There are two important hypotheses which mean the result holds. Firstly the system you are interested in is *elliptic*. Secondly there is *no kernel*, i.e. no solutions to the homogeneous equation. I will explain the result in this sort of generality (linear elliptic systems over a compact Riemannian manifold) since it's less notation and no more extra thought.


So, let $(M,g)$ be a compact Riemannian manifold with vector bundles $E, F \to M$ and let $L$ be a linear differential operator taking sections of $E$ to sections of $F$. In your case, $M=S^2$ is the shpere, $E=T^\*S^2$ is the cotangent bundle and $F$ is the sum of two real line bundles: $F = \underline{\mathbb{R}} \oplus \Lambda^2T^\*S^2$. Your operator is:
$$
L(\alpha) = (\mathop{div}(\alpha), \mathop{curl}(\alpha))
$$


We aassume that $L$ is elliptic. This means that the principal symbol in any given cotangent direction is an isomorphism $E \to F$. This is the case for your $L$ because the symbol in the direction $\beta$ is given by
$$
\sigma\_L(\beta)(\alpha) = ( g(\alpha,\beta), \beta \wedge \alpha)
$$
(To see that $\sigma\_L$ is an isomorphism it's enough to check it has no kernel.)


The first estimate I will explain is that for any elliptic $L$, any $k \in \mathbb{N}$ and any $p \in (1,\infty)$, there is a constant $C$ (which depends on $L,k,p$) such that for any section $s$ of $E$,
$$
| s |\_{L^p\_{k+r}} \leq C \left( | L(s)|\_{L^p\_k} + |s |\_{L^p}\right)
$$
Here $r$ is the order of $L$ (so $r=1$ in your cases). Meanwhile, $L^p\_k$ means the Sobolev norm given by taking the sum of the $L^p$ norms of derivatives up to order $k$, all defined using the Riemannian metric $g$. We also need to fix a choice of fibrewise innerproduct in both $E$ and $F$,as well as connections in the bundles. (Your question only asks about $k=1,2$.)


To prove this estimate, use a partition of unity to write $s = \sum \phi\_i s$ where each $\phi\_i$ is supported in a small coordinate chart. Suppose moreover that $E$ and $F$ are both trivial over each of these coordinate patches. Now on any single coordinate chart, $\Omega\_i$, you can treat everything as if it were taking place on vector valued functions on $\mathbb{R}^2$. Apply the classical Calderon-Zygmund inequality to obtain:
$$
| \phi\_i s |\_{W^p\_{k+r}(\Omega\_i)} \leq C| L(\phi\_i s)|\_{W^p\_k(\Omega\_i)}
$$
where here $W^p\_k(\Omega\_i)$ means using the Euclidean $(p,k)$-Sobolev norm for vector valued functions on $\Omega\_i \subset \mathbb{R}^n$. *It is important here that $\phi\_i s$ is compactly supported in $\Omega\_i$.*


To get a global inequality over the whole of $M$, we need to do two things: Firstly, we need to compare the $W^p\_k(\Omega\_i)$ norm and $L^p\_k(\Omega\_i)$ norm for sections supported in $\Omega\_i$. It turns out that there is a constant $K>0$ such that for any section of $E$ or $F$ supported in $\Omega\_i$,
$$
\frac{1}{K} | s |\_{L^p\_k(\Omega\_i)} \leq | s|\_{W^p\_k(\Omega\_i)} \leq K | s|\_{L^p\_k(\Omega\_i)}.
$$
The constant $K$ will depend on $\Omega\_i$, because we need to compare $g$ to the Euclidean metric on each chart $\Omega\_i$ (and also our connections on $E$ and $F$ to the trivial connection). When $M$ is compact, there are only finitely many charts, so we can use the same $K$ for all of them. (When $M$ is not compact, one needs to assume some uniform geometric bounds here.)


From here we see that for any section $s$ at all,
$$
| s|\_{L^p\_{k+r}(M)} 
\leq K \sum | \phi\_i s|\_{W^p\_{k+r}(\Omega\_i)} \leq CK^2 \sum | L(\phi\_i s)|\_{W^p\_k(\Omega\_i)}
$$
where the second inequality is Calderon-Zygmund in $\Omega\_i$.


Next, we need to relate $L(\phi s)$ to $\phi\_i L(s)$. We do this via the Leibniz rule. The difference $L(\phi\_i s) - \phi\_i L(s)$ is an expression in which we see terms with at most $r-1$ derivatives of $s$. This means that (with a little more work),
$$
| L(\phi\_i s) |\_{W^p\_k(\Omega\_i)} \leq B \left(| \phi\_i L(s)|\_{W^p\_k(\Omega\_i)} + | s |\_{W^p\_{r+k-1}(\Omega\_i)}\right)
$$
Multiplication by a smooth function is a bounded map on Sobolev spaces and so
$$
| \phi\_i L(s)|\_{W^p\_k(\Omega\_i)} \leq A |L(s)|\_{W^p\_k(\Omega\_i)}
$$
Putting this together we find that
$$
| s|\_{L^p\_{k+r}(M)} \leq D \sum \left(|L(s)|\_{L^p\_k(\Omega\_i)} + |s|\_{L^p\_{k+r-1}(\Omega\_i)}\right)
$$
where $D$ combines all our previous constants $C,K,A,B$.


We also choose our coordinate charts to be *locally finite*: any point of $M$ is in at most $m$ of the $\Omega\_i$. This means that
$$
\sum |L(s)|\_{L^p\_k}(\Omega\_i) \leq m |L(s)|\_{L^p\_k(M)}
$$
and similarly for the $L^p\_{k+r-1}$ term. So we conclude that
$$
| s|\_{L^p\_{k+r}(M)} \leq C \left( |L(s)|\_{L^p\_k(M)} +|s|\_{L^p\_{k+r-1}}\right)
$$
for some new constant $C$. We now get the estimate we really want (with $L^p\_{k+r-1}$ replaced by $L^p$ in the last term) by induction on $k$. (When $r=1$ the base of the induction is simple enough, but when $r>1$ it needs a little more thought, which I leave as an exercise - code for I didn't have time to think about this bit very much!)


This is almost the estimate you want in your original quesiton - which asked about the cases $k=1,2$ for $L = (\mathop{div}, \mathop{curl})$ - *except there is an additional $L^p$ norm of $s$ on the right-hand side.* To get rid of this we need an additional hypothesis: we assume that $L$ has no kernel. In your case this is true because the first Betti number of $S^2$ vanishes. (The estimate would be FALSE on the torus for example. You can see this by taking, in your notation, $\alpha$ to be a non-zero parallel 1-form. Then $\mathop{div}(\alpha)=0=\mathop{curl}(\alpha)$.)


With this additional assumption, let us prove that there exists $C$ for which
$$
|s|\_{L^p(M)} \leq C|L(s)|\_{L^p\_k(M)}
$$
If not then, by contradiction, there is a sequence $s\_j$ of sections of $E$ with $|s\_j|\_{L^p} =1$ and with $L(s\_j) \to 0$ in $L^p\_k(M)$. By the estimate we have just proved, $s\_k$ is bounded in $L^{p}\_{k+r}$. By Rellich's Lemma, a subsequence converges in $L^p\_{k+r-1}$. The limit $s$ must be a solution of $L(s)=0$ and so must vanish, but this contradicts the fact that $|s|\_{L^p}=1$. This means that, when $\mathrm{Ker}(L)=0$, we can absorb the $|s|\_{L^p}$ term on the right-hand side in our previous estimate into the $|L(s)|\_{L^p\_k}$ term.


This completes the proof that when $L$ is elliptic and has no kernel, there is a constant $C$ (depending on $k,p$) such that
$$
|s|\_{L^p\_{k+r}(M)} \leq C |L(s)|\_{L^p\_{k}(M)}
$$
This is what you asked for, when you set $k=1,2$ and $L = (\mathop{div},\mathop{curl})$ acting on 1-forms over the 2-sphere.


