Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała

Colloquium Mathematicae, Tome 144 (2016), p. 137-148 / Harvested from The Polish Digital Mathematics Library

Résumé

Let 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 (*), where $^{∨}$ and $^{∧}$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . 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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Publié le : 2016-01-01

Zbl 06602776

EUDML-ID : urn:eudml:doc:286085