  We prove uniform boundedness of certain boundary representations on
appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov
hyperbolic groups. These are closed subspaces of $L^p$ and in particular
Hilbert spaces in the case $p=2$.
  This construction allows us, for an appropriate choice of $p$, to approximate
the trivial representation through uniformly bounded representations. This
phenomenon does not have analogue in the setting of isometric representations
whenever the hyperbolic group considered has the Property (T).
  The key is the introduction of a notion of metrically conformal operator on a
metric space endowed with a conformal structure \`{a} la Mineyev and a metric
analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform.
