A functional calculus of pseudodifferential operators on unimodular Lie groups

Abstract

An algebra of pseudodifferential operators on a unimodular Lie group is constructed. It is defined by uniform estimates of the local symbols and by conditions on the decrease of the kernel outside the diagonal, formulated in terms of the weight function, having an intermediate growth between the volume function and a standard exponential. The decrease of the Green function is described, complex powers of elliptic operators of the considered classes are constructed, a meromorphic continuation of the kernels of complex powers is described, from where, in particular, the asymptotic behavior of the spectral function of operators of the considered class is obtained.