Lower bounds for operators on graded Lie groups

TL;DR

This paper develops a symbolic pseudo-differential calculus on graded Lie groups and applies it to establish lower bounds for positive Rockland operators, enhancing understanding of their hypoelliptic properties.

Contribution

It introduces a new symbolic calculus on graded Lie groups and derives sharp lower bounds for Rockland operators with variable coefficients.

Findings

Established a symbolic pseudo-differential calculus on graded Lie groups

Proved a sharp Garding inequality for these operators

Derived lower bounds and hypoellipticity results for Rockland operators

Abstract

In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.