Leibniz’s rule on two-step nilpotent Lie groups
Volume 145 / 2016
Abstract
Let $\mathfrak {g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell–Hausdorff multiplication. This allows us to define a generalized multiplication $f \mathbin {\#} g = (f^{\vee } * g^{\vee })^{\wedge }$ 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 $\mathfrak {g}^{*}$ and $*$ is the convolution on the group $\mathfrak {g}$.
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 $f \mathbin {\#} g$ for some classes of symbols.
We find a formula for $D^{\alpha }(f \mathbin {\#} g)$ for Schwartz functions $f,g$ in the case of two-step nilpotent Lie groups, which 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, which includes the usual classes of symbols. In the case of the Abelian group $\mathbb {R}^{d}$ we have $f \mathbin {\#} g = fg$, so $D^{\alpha }(f \mathbin {\#} g)$ is given by the Leibniz rule.
