The semigroup and the inverse of the Laplacian on the Heisenberg group

Abstract

By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators L_Ï„,Ï„ ∈ â„ \ {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by L_Ï„, and the inverse of L_Ï„. Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.