  By using commutator methods, we show uniform resolvent estimates and obtain
globally smooth operators for self-adjoint injective homogeneous operators $H$
on graded groups, including Rockland operators, sublaplacians and many others.
Left or right invariance is not required. Typically the globally smooth
operator has the form $T=V|H|^{1/2}$, where $V$ only depends on the homogeneous
structure of the group through Sobolev spaces, the homogeneous dimension and
the minimal and maximal dilation weights. For stratified groups improvements
are obtained, by using a Hardy-type inequality. Some of the results involve
refined estimates in terms of real interpolation spaces and are valid in an
abstract setting. Even for the commutative group $\R^N$ some new classes of
operators are treated.
