Symbolic calculus and convolution semigroups of measures on the Heisenberg group

Abstract

Let $P$ be a generalized laplacian on $R^{2n+1}$. It is known that $P$ is the generating functional of semigroups of measures $\mu_{t}$ on the Heisenberg group $H^{n}$ and $\nu_{t}$ on the Abelian group $R^{2n+1}$. Under some smoothness and growth conditions on the functional $P$ expressed in terms of its Abelian Fourier transform $\widehat{P}$ we show that the semigroup $\mu_{t}$ is a kind of perturbation of the semigroup $\nu_{t}$. More precisely, we give pointwise estimates for the difference of the densities of the measures $\mu_{t}$ and $\nu_{t}$. As a consequence we get a description of the asymptotic behavior at the origin or pointwise estimates for the densities of the semigroup of measures on the Heisenberg group which is an analogue (via generating functional) of the symmetrized gamma (gamma-variance) semigroup on $R^{2n+1}$. The main tool is a symbolic calculus for convolution operators on the Heisenberg group.