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Leibniz’s rule on two-step nilpotent Lie groups

Volume 145 / 2016

Krystian Bekała Colloquium Mathematicum 145 (2016), 137-148 MSC: Primary 22E25; Secondary 22E15. DOI: 10.4064/cm6573-10-2015 Published online: 3 June 2016


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.


  • Krystian BekałaInstitute of Mathematics
    University of Wrocław
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland

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