Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations

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 Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's inequality due to Gårding [26].

31 Pages

References in corpus (3)

• Fractional Quantum Mechanics
• Lower bounds for operators on graded Lie groups
• Integration by parts for nonsymmetric fractional-order operators on a halfspace