Leibniz's rule on two-step nilpotent Lie groups

TL;DR

This paper derives a Leibniz rule for the product of functions on two-step nilpotent Lie groups, extending symbolic calculus and pseudodifferential operator analysis in this setting.

Contribution

It provides a formula for derivatives of the generalized product of functions on two-step nilpotent Lie groups, extending the symbolic calculus framework.

Findings

Derived a Leibniz rule for the product on two-step nilpotent Lie groups.

Extended the formula to distributions acting by convolution.

Connected the results to classical Leibniz rule in the Abelian case.

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 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 H\"ormander. The idea of such a calculus consists in describing the product $f \# g$ for some classes of symbols. We find a formula for $D^{\alpha}(f \# g)$ 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 $D^{\alpha}(f \# g)$ is given by the Leibniz rule.