Analytic functional calculus and G\r{a}rding inequality on graded Lie groups with applications to diffusion equations

TL;DR

This paper develops a global complex functional calculus on graded Lie groups to analyze diffusion equations with hypoelliptic pseudo-differential operators, leading to new energy estimates and a G {a}rding inequality that generalizes classical results.

Contribution

It introduces a novel global functional calculus on graded Lie groups, extending classical results to hypoelliptic operators and deriving a generalized G {a}rding inequality.

Findings

Established a global complex functional calculus on graded Lie groups.

Recovered the standard calculus for Euclidean H"ormander classes.

Proved a G {a}rding inequality applicable to all graded Lie groups.

Abstract

In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(\rho,\delta)$-Euclidean H\"ormander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the G\r{a}rding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's inequality due to G\r{a}rding [26].