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

TL;DR

This paper introduces a novel framework for transferring uniform hypoelliptic properties from local to global settings on graded Lie groups, using a 'freeze-unfreeze' strategy that avoids pseudodifferential calculus.

Contribution

It develops a new method to lift local hypoelliptic estimates to global ones on graded Lie groups, and proves self-adjointness of symmetric hypoelliptic operators.

Findings

Established a mechanism for global hypoelliptic estimates

Extended the 'freeze-unfreeze' technique to hypoelliptic operators

Proved self-adjointness of symmetric hypoelliptic operators

Abstract

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.