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Leibniz's rule on two-step nilpotent Lie groups

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Let 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 $f # g = (f^{∨} ∗ g^{∨})^{∧}$ of two functions in the Schwartz class 𝓢(𝔤*), where $^{∨}$ and $^{∧}$ are the Abelian Fourier transforms on the Lie algebra 𝔤 and on the dual 𝔤* and ∗ is the convolution on the group 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, which includes the Heisenberg group. We extend this formula to the class of functions f,g such that $f^{∨}, g^{∨}$ are certain distributions acting by convolution on the Lie group, which includes the 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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  • Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm6573-10-2015
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