  We study quantum oscillator lattice systems with disorder, in arbitrary
dimension, requiring only partial localization of the associated effective
one-particle Hamiltonian. This leads to a many-body localized regime of excited
states with arbitrarily large energy density. We prove zero-velocity
Lieb-Robinson bounds for the dynamics of Weyl operators as well as for position
and momentum operators restricted to this regime. Dynamical localization is
also shown in the form of quasi-locality of the time evolution of local Weyl
operators and through exponential clustering of the dynamic correlations of
states with localized excitations.
