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This might be a bit naive question, but I am really curious about this.

As I know, the original Yau's proof, and the proofs in many literatures use the Hölder space theory to prove Calabi's conjecture. Maybe, because we need a strong solution of the complex Monge-Ampère equation, it is natural to consider Hölder spaces. However, if there might be suitable regularity results relevant to the problem, it would be possible to solve the problem by using Sobolev spaces or some similar weak solution method.

I do not know any source in this direction, but someone might think about that, or there might be some obstacle to use the Sobolev space theory.

Thanks!

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    $\begingroup$ The proof uses both Hölder and Sobolev norms. See, e.g., the exposition of D. Joyce in his book Riemannian holonomy groups and calibrated geometry. $\endgroup$ – Spenser Feb 28 at 18:33
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    $\begingroup$ @Spenser In my understanding, the part using the Sobolev norms is for the $C^0$-estimate. In this part, the Sobolev inequality is used, so in some sense, I agree that Sobolev space theory is used in the proof. However, my intention for the question is more like about some use of weak solutions: to show existence of some weak solution, and to guarantee it is a strong solution by some regularity results. $\endgroup$ – Yongmin Park Mar 1 at 7:13

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