[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, (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.