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[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.

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  • $\begingroup$ Can you remind us of the name of the paper or give a link? $\endgroup$
    – David Roberts ♦
    Commented Mar 16, 2022 at 7:11
  • $\begingroup$ sorry, On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I* ,jasonpayne.webs.com/Math5339/… $\endgroup$
    – Elio Li
    Commented Mar 16, 2022 at 10:22

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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.

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