  We consider asymptotically hyperbolic manifolds whose metrics have
Sobolev-class regularity, and introduce several technical tools for studying
PDEs on such manifolds. Our results employ two novel families of function
spaces suitable for metrics potentially having a large amount of interior
differentiability, but whose Sobolev regularity implies only a H\"older
continuous conformal structure at infinity. We establish Fredholm theorems for
elliptic operators arising from metrics in these families.
  To demonstrate the utility of our methods, we solve the Yamabe problem in
this category. As a special case, we show that the asymptotically hyperbolic
Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal
compactification, with $p$ greater than the dimension of the manifold.
