Equivalence between two Sobolev norms on manifolds

Question

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.

1. Use pseudo-differential operators on $M$: $$\|f\|_{H^{s,p}(M)}=\|(I-\Delta_g)^{s/2}f\|_{L^p(M)}.$$

2. Use local coordinate and Fourier transform in $\mathbb{R}^n$

$$\|f\|_{W^{s,p}(M)}=\sum_{\nu}\|D^sf_\nu\|_{L^p(\mathbb{R}^n)},$$ Here $f_\nu=(\phi_\nu\cdot f)\circ \kappa_\nu^{-1}$, where $\{\phi_\nu\}$ is a partition of unity subordinate to a finite covering $M=\cup \Omega_\nu$, $\kappa_\nu:\Omega_\nu\to \tilde \Omega_\nu\subset \mathbb{R}^n$ is the coordinate map, and $D^sf_\nu$ is defined by the Fourier transform in $\mathbb{R}^n$ $$(D^s g)^\wedge(\xi)=(1+|\xi|^2)^{s/2}\hat g(\xi).$$

It is well-known that these two norms are equivalent when $p=2$. It can be proved by using the $L^2$-boundedness of zero-order pseudo-differential operators. However, I cannot find any reference on the equivalence for $p\ne2$. Any help will be appreciated.

Answer 1

There are in fact more than two ways to define these norms, and finding good references for their equivalence is difficult. The statement you search for above is correct, provided you assume additionally the charts satisfy the property that the closure of $\Omega_\nu$ is contained in an open set $U_\nu$ such that the chart can be extended to $U_\nu$. Such an atlas always exists, but if you choose a chart without this additional property, bad things can happen: in particular your definition 2.) depends on the charts that you take.

Answer 2

A partial reference that could partially help to you is Bernd Ammann, Alexandru Ionescu, Victor Nistor Sobolev spaces on Lie manifolds and regularity for polyhedral domains Doc. Math. 11 161-206 (2006) https://www.math.uni-bielefeld.de/documenta/vol-11/07.html

The main target of our reference is a class of non-compact manifolds, but essentially everything in there also applies to compact manifolds as well, by considering the special case that "the boundary at infinity" is the empty set. There it is shown that three possible definitions of H^{s,p}, namely Def. 3.1, Def. 3.2 and Def. 3.8 all coincide. It is then also evident that these definitions conincide to the (repaired) definition 2. above. In order to get what you want, it remains to add that $(I-\Delta)^{s/2}$ is an isomorphism from $H^{s,p}$ to $L^p$, provided $1<p<\infty$. The continiuty of this map is proven in the above reference for positive $s\in 2\mathbb{Z}$ in Corollary 3.11. Yes, this only partially solves your problem, but at least it is a first step. With similar techniques and using local properties for pseudo-differential operators, e.g. Taylor, Pseudodifferential operators, Princeton Math. Series, Chapter I, §6, such statements may be obtained.