I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe Problem by Lee and Parker [LP].

Some definitions:

$$W^{k,p}_\delta =\{f\in L^p_{loc}(\mathbb{R}^n) \, \vert \, \sum_{i\leq k} \Vert \nabla^i f\Vert_{p,\delta-i} <+\infty \}$$ where

$$\Vert f\Vert_{p,\delta}^p= \int_{\mathbb{R}^n} \vert f\vert^p \sigma^{-p\delta-n}\ dx,$$ where $\sigma= 1+\vert x\vert$.

Exceptionnal values are the order of decreasing of harmonic function at infinity, namely $\mathbb{Z}\setminus\{-1, \dots,-n-1\}$.

A metric is assumptotically flat if there exists a compact $K$ and a chart $\phi :M\setminus K \rightarrow \mathbb{R}^n\setminus B_1$ such that $\phi_*(g)$ is uniformly equivalent to $\delta$ and $\phi_*(g) -\delta \in W^{1,q}_{-\tau}$ for some $\tau \geq 0$ and $q>n$, the decrease rate of the metric.

Both state that if $\tau$ is positive and $1<p\leq q$ then $\Delta_g:W^{2,p}_\delta \rightarrow L^{p}_{\delta-2}$ is surjective if $\delta>2-n$ and non-exceptional, and injective if $\delta<0$ and non-exceptional. As consequence of the Fredhom theory and explicit knowledge of the Kernel of the flat laplacian. I totally agree with this.

Then in order to prove the existence of harmonic coordinates at infinity , their statements diverge a bit. They both remark, as consequence of the definition of asymptotic flatness, that $$\Delta_g x^i\in L^{p}_{-1-\tau},$$ which I agree too. But Bartnik say it is enough to solve $$\Delta_g v_i =\Delta_g x^i$$ in $W^{1,p}_{1-\tau}$, but we should have $1-\tau >2-n$, i.e. $\tau < n-1$ , which is specify nowhere. And Lee and Parker make almost the same for $n\geq 4$ but replace $1-\tau$ by $1-\tau +\epsilon$ when $n=3$, with $\epsilon$ in order to have $1-\tau +\epsilon>n-2$ which seems more reasonable but I don't see why they don't face this issue in dimension $\geq4$.

I would appreciate any enlightenment or clearer reference about this. Thx