  The properties of the electron-phonon interaction in the presence of a
sizable electronic repulsion at finite doping are studied by investigating the
metallic phase of the Hubbard-Holstein model with Dynamical Mean Field Theory.
Analyzing the quasiparticle weight at finite doping, we find that a large
Coulomb repulsion reduces the effect of electron-phonon coupling at low-energy,
while this reduction is not present at high energy. The renormalization of the
electron-phonon coupling induced by the Hubbard repul sion depends in a
surprisingly strong and non-trivial way on the phonon frequency. Our results
suggest that phonon might affect differently high-energy and low-energy
properties and this, together with the effect of phonon dynamics, should be
carefully taken into account when the effects of the electron-phonon
interaction in a strongly correlated system, like the superconducting cuprates,
are discussed.
