Uniformity of Maximal Hypoellipticity on Graded Lie Groups: From Pointwise to Global

On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old “freeze-unfreeze” strategy into the hypoelliptic setting, and offers a transparent and flexible framework for lifting symbol-level hypoelliptic properties to global elliptic estimates, without relying on pseudodifferential calculus. In addition, we prove that symmetric operators of hypoelliptic type on a graded Lie group are self-adjoint.