I am studying the existence of harmonic coordinates at infinity on on 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]. Both state that if the decrease rate($\tau$) of the metric is positive 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 I totally agree with. Then in order to prove the existence of harmonic coordinates at one end, their statements diverge a bit. They both remark that $$\Delta_g x^i\in L^{p}_{-1-\tau},$$ which I agree to. But Bartnik say it is enough to solve $$\Delta_g v_i =\Delta_g xi$$ in $W^{1,p}_{1-\tau}$, but we should have $1-\tau >2-n$, i.e. $\tau < n-1$ which specify nowhere. And Lee&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