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@DeaneYang, thank you. It is so difficult to study the book of Aubin. I really need to learn about Sobolev spaces on manifolds. Do you know a "better" book to study about it? Thanks.
I will write the theorem 1.53 here and will try to explain the notations. However, we need a lot of definitions and notations of this book to understand my doubt. Unfortunatly, the own notation is confusing. Anyway, I will write it later. Thank you.
I cited the book because he explain that this is a consequence of a theorem. This inequality is a step of a proof, which he uses another result to obtain it. If I had wrote here, would be confusing. The reference is necessary to understand it.
Connor, thank you for your patience and attention. I didn't understand some steps of your proof. First, observe that I have no information about the signal of $f$, and you assumed $g>0$. Second, could you explain me how to obtain the estimate $0\leq u\leq C(M)x_{n}$ using the barrier function? Finally, can you re-explain the steps $3$ and $4$? Thank you very much.
