I've been reading a paper "Schatten classes on compact manifolds: Kernel conditions" by Delgado and Ruzhansky (https://arxiv.org/abs/1403.6158).
In the paper, the authors define the Sobolev space $H^\mu(M)$ on a closed manifold $M$ in the following way:
$H^\mu(M)$ is the space of all distributions $f\in D'(M)$ such that $(I+P)^{\frac{\mu}{\nu}}f\in L^2(M)$ where $P$ is a positive elliptic operator of order $\nu$. Also, this characterisation is independent of the choice of operator $P$.
It is difficult to understand the paper without understanding of the above definition. But I only know basic definition of Sobolev space for $\mathbb{R}^n$ (I learned that from Folland's real analysis book long time ago). Also, I have no background on the elliptic operator for manifolds (I only know that Laplace-Beltrami operator is elliptic).
So, I'm looking for any good books/papers explaining the above definition of Sobolev spaces on manifolds. I tried some googleing but the books I found were not using the above definition.