[This question is looking at the paper


* Yau, S.-T., *On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I*, Comm. Pure Appl. Math., **31** (1978) 339-411, doi:[10.1002/cpa.3160310304](https://doi.org/10.1002/cpa.3160310304), ([pdf](https://jasonpayne.webs.com/Math5339/On%20the%20Ricci%20Curvature%20of%20a%20Compact%20Kahler%20Manifold%20and%20the%20Complex%20Monge-Ampere%20Equation%20I,%20S.T.%20Yau.pdf))]


My problem arises from (2.43)
$$
\Delta \varphi=f
$$
where
$$
-m \leqq f \leqq C\_{1} \exp \{C \sup \varphi\} \exp \left\{-\inf \_{M} \varphi\right\}.
$$


The paper already has a estimation of $\sup\varphi$ then the paper gets:
$$
\sup \_{M}|\nabla \varphi| \leqq C\_{6}\left(\exp \{-C \inf \varphi\}+\int\_{M}|\varphi|\right)
$$
How does this step use the Schauder estimate? Why is there a $\int\_{M}|\varphi|$ on the right handside, should it be $\sup|\varphi|$?


I'm so confused about this step, thanks for helping.



A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture
There are many non-equivalent formulations of the Schauder estimates; the version you are suggesting is the most common. For the version he needs, Yau gives a precise page & theorem reference (p.156, formula 5.5.23) to Morrey's book. In the case of the Euclidean Laplacian you can also see p.69-70 of Gilbarg & Trudinger.


