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- W2395341010 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 Hormander. 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." @default.
- W2395341010 created "2016-06-24" @default.
- W2395341010 creator A5054457737 @default.
- W2395341010 date "2015-01-30" @default.
- W2395341010 modified "2023-09-27" @default.
- W2395341010 title "Leibniz's rule on two-step nilpotent Lie groups" @default.
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