# dg.differential geometry – Two questions on some operator showing within the proof of Calabi’s surmise Answer

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## dg.differential geometry – Two questions on some operator showing within the proof of Calabi’s surmise

I’m erudition in regards to the proof of Calabi’s surmise. The proof makes use of the continuity system to decipher the complicated Monge-Ampère equation, and within the proof, I necessity to show that the next operator is of a minimum of $$C^1$$ (between some Banach areas) to employ the inverse duty theorem:

$$mathcal{M}(psi) = frac{(omega + sqrt{-1} partial barrier{partial} psi)^n}{omega^n},$$
the place $$omega$$ is a hard and fast Kähler figure on a compact Kähler manifold $$M$$. My questions are the next:

1. Many authors employ some Hölder areas because the province and the goal of this operator, for instance, $$C^{3, alpha}$$ and $$C^{1, alpha}$$. Is there any intuition for such a altenative? It appears that it’s for utilizing some Schauder idea, however what if one employ Sobolev areas or every other duty areas?
2. I can ascertain (not display formally) the Fréchet spinoff of this operator which is given by
$$D mathcal{M}(psi)(eta) = frac{n sqrt{-1} partial barrier{partial} eta wedge (omega + sqrt{-1} partial barrier{partial} psi)^{n-1}}{omega^n}.$$
However, I can not show the operator $$mathcal{M}$$ is of $$C^1$$ (or plane $$C^okay$$) certainly. I cerebrate that to ensure that $$mathcal{M}$$ is of $$C^1$$, it is sufficient to show that the next operators are steady:
commence{align} (alpha, beta) &mapsto alpha wedge beta psi &mapsto sqrt{-1} partial barrier{partial} psi, aim{align}
however the issue is being steady in what sense. Should I deem the $$L^2$$-spaces of differential types, or one other duty areas? I can not design out what I necessity to appraise for this job.

Thanks!

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