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.