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 pρ, δq-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the G̊arding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953’s inequality due to G̊arding [26].

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