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By using commutator methods, we show uniform resolvent estimates and obtainglobally 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 smoothoperator has the form $T=V|H|^{1/2}$, where $V$ only depends on the homogeneousstructure of the group through Sobolev spaces, the homogeneous dimension andthe minimal and maximal dilation weights. For stratified groups improvementsare obtained, by using a Hardy-type inequality. Some of the results involverefined estimates in terms of real interpolation spaces and are valid in anabstract setting. Even for the commutative group $\R^N$ some new classes ofoperators are treated.

Resolvent Estimates and Smoothing for Homogeneous Operators on Graded Lie Groups