Symbolic calculus and convolution semigroups of measures on the Heisenberg group

Krystian Bekała

Published 2015-01-30Version 1

Let $P$ be a symmetric generalised laplacian on $R^{2n+1}$. It is known that $P$ generates semigroups of measures $\mu_{t}$ on the Heisenberg group $H^{n}$ and $\nu_{t}$ on the Abelian group $R^{2n+1}$. Recall that the underlying manifold of the Heisenberg group is $R^{2n+1}$. Suppose that the negative defined function $\psi(\xi)=-\hat{P}(\xi)$ satisfies some weight conditions and $|D^{\alpha}\psi(\xi)| \leq c_{\alpha}\psi(\xi)(1+\|\xi\|)^{-|\alpha|}, \xi \in R^{2n+1}.$ 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 between the densities of $\mu_{t}$ and $\nu_{t}$ and we show that it is small with respect to $t$ and $x$. As a consequence we get a description of the asymptotic behaviour at origin of densities of a semigroup of measures which is analogon of the symmetrized gamma (gamma-variance) semigroup on the Heisenberg group.