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

plane geometry - Does the intersection of two moving curves sweep out a continuous line?

Hello dear visitor to our network We will offer you a solution to this question 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 ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

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.

we will offer you the solution to 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 question via our network which brings all the answers from multiple reliable sources.