Let [Special characters omitted.] be a nilpotent, stratified homogeneous group, and let X 1 , · · ·, X l be left invariant vector fields generating the Lie algebra [Special characters omitted.] associated to [Special characters omitted.] . In [13] G. Lu proved a Fefferman-Phong type inequality for degenerate vector fields and stated that various operators associated with the sub-Laplacian [Special characters omitted.] plus a nonnegative potential are L p bounded on the homogeneous group [Special characters omitted.] when V ( x ) is a nonnegative group polynomial on [Special characters omitted.] or satisfies a certain Reverse Hölder inequality in the metric space ([Special characters omitted.] , ρ) defined by the homogeneous norms. In this paper, we provided details of proof of theorems stated in [13]. Furthermore, we extended the results to higher order subelliptic operators [Special characters omitted.] and [Special characters omitted.] when V is a nonnegative polynomial. We obtained the L p boundedness for various operators related to the two operators above. We also gave the fundamental solution estimates for [Special characters omitted.] and proved that the fundamental solution to [Special characters omitted.] are differentiable away from the pole and behaves like that of [Special characters omitted.] for ρ ( x, y ) < m ( y, V ) -1 while decays faster than any negative power of ρ ( x, y ) for ρ ( x, y ) > m ( y, V ) -1 . Finally, to get the fundamental solution estimates to [Special characters omitted.] , we proved Caccioppoli Inequality and Mean-Value Inequality for the equation [Special characters omitted.] u ( x ) = 0
