I'm reading the paper ON CHERN-YAMABE PROBLEM

in this paper, $\Delta^{C h} f=\Delta_{d} f+(d f, \theta)_{\omega}$, after getting the uniform $L_{\infty}$ bound of the constructed sequence $f_{t_{n}}$, the problem is the regularity of $$L_{n} f_{t_{n}}=\lambda \exp \left(2 f_{t_{n}} / n\right)$$ where $$L_{n} f:=\Delta^{C h} f+t_{n} S^{C h}(\omega)+\lambda\left(1-t_{n}\right)$$ the paper says the right-hand side $\lambda \exp \left(2 f_{t_{n}} / n\right)$ of the equation. Then, by iterating the Calderon-Zygmund inequality and using Sobolev embeddings, we find, let us say, an a-priori $\mathcal{C}^{3}$ uniform bound. Then using arzela-ascoli we get the converging sequence.

How the Calderon-Zygmund inequality is used here? The $(d f, \theta)_{\omega}$ here may be seen as $h \cdot\nabla f$,if we don't have the $(d f, \theta)_{\omega}$ term, then we just iterate like $\lambda \exp \left(2 f_{t_{n}} / n\right) \in H^{k,p}$ then $f_{t_{n}} \in H^{k+2,p}$...

But if there is a $(d f, \theta)_{\omega}$ term I don't know how to deal with the iteration. I google the Calderon-Zygmund inequality and found many types but I didn't find how they are used to my problem.



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