The Sobolev space on Euclidian space $W^{k,p}(\Omega)$ is well-known, where $\Omega$ is a subset in $\mathbb{R}^{d}$. Then, how does the Sobolev space on the compact Riemannian manifold $W^{k,p}(M)$ be defined? I have searched a variety of reference literatures but the definition for $H^{k}(M)$, i.e., the case of $p=2$ is only written in those literatures and not written for general $p$. By the way, is the way via the Fourier transform general in the definition of $H^{k}(M)$?

If you know the definition for $W^{k,p}(M)$ or some reference literatures, please let me know. Also I want to know some properties, for example, Sobolev embedding etc.

In addition, I'm also interested in the Sobolve space $W^{k,p}(X)$, where $X$ is (complete?) metric space. Because I don't know at all for this, I'm glad if you give some information.

Thank you.


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