# fa.functional analysis – Calderon-Zygmund in solving $\Delta u + (d f, \theta)_{\omega}=e^{u}$ when we have the $L_{\infty}$ estimation of $u$ on complex compact manifold Answer

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## fa.functional analysis – Calderon-Zygmund in solving $\Delta u + (d f, \theta)_{\omega}=e^{u}$ when we have the $L_{\infty}$ estimation of $u$ on complex compact manifold

I’m reading the paper ON CHERN-YAMABE PROBLEM

in this paper, $$\Delta^{C h} f=\Delta_{d} f+(df, \theta)_{\omega}$$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 $$(df,\theta)_{\omega}$$ here may be seen as $$h \cdot\nabla f$$，if we don’t have the $$(df,\theta)_{\omega}$$ 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 $$(df,\theta)_{\omega}$$ 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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