Krystian Bekała | Searcharxiv

arXiv2017

Symbolic calculus and convolution semigroups of measures on the Heisenberg group

Let $P$ be a generalized laplacian on $R^{2n+1}$. It is known that $P$ is the generating functional of semigroups of measures $μ_{t}$ on the Heisenberg group $H^{n}$ and $ν_{t}$ 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 $μ_{t}$ is a kind of perturbation of the semigroup $ν_{t}$. More precisely, we give pointwise estimates for the difference of the densities of the measures $μ_{t}$ and $ν_{t}$. 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.

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arXiv2016

Leibniz's rule on two-step nilpotent Lie groups

Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two functions in the Schwartz class $\mathcal{S}(\mathfrak{g}^{*})$, where $\vee$ and $\wedge$ are the Abelian Fourier transforms on the Lie algebra $\mathfrak{g}$ and on the dual $\mathfrak{g}^{*}$. In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order generalization of the Weyl calculus for pseudodifferential operators of Hörmander. The idea of such a calculus consists in describing the product $f \# g$ for some classes of symbols. We find a formula for $D^α(f \# g)$ for Schwartz functions $f,g$ in the case of two-step nilpotent Lie groups, that includes the Heisenberg group. We extend this formula to the class of functions $f,g$ such that $f^{\vee}, g^{\vee}$ are certain distributions acting by convolution on the Lie group, that includes usual classes of symbols. In the case of the Abelian group $R^{d}$ we have $f \# g = fg$, so $D^α(f \# g)$ is given by the Leibniz rule.

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