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Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation: $(\omega_0 +i \partial \bar \partial \varphi)^...

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...

I want to know the space of functions where the following quantity is uniformly bounded from above $$\int_{K} (\Delta u)^j d\lambda< C,$$ where K is a compact and j is an integer number greater ...