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

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].