Maximal Hypoellipticity for Left-Invariant Differential Operators on Lie Groups

Given a differential operator defined in terms of left-invariant vector fields on a Lie group, we prove that the local condition defining maximal hypoellipticity is equivalent to a global estimate if the operator is left invariant. As a consequence, we characterize the domain of the closure in $L^p$ of certain maximal hypoelliptic differential operators on test functions on Lie groups.