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

Aparajita Dasgupta; M.W. Wong

doi:10.4067/S0719-06462010000300006

Authors

Aparajita Dasgupta Department of Mathematics, Indian Institute of Science, Bangalore–560012, India.
M.W. Wong Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada.

DOI:

https://doi.org/10.4067/S0719-06462010000300006

Keywords:

Heisenberg group, Laplacian, parametrized partial differential operators, Hermite functions, Fourier-Wigner transforms, heat equation, one parameter semigroup, inverse of Laplacian, Sobolev spaces

Abstract

By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators L_Ï„,Ï„ âˆˆ 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.