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Leibniz's rule on two-step nilpotent Lie groups
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bekala, Krystian |
| Copyright Year | 2015 |
| 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. |
| Starting Page | 137 |
| Ending Page | 148 |
| Page Count | 12 |
| File Format | PDF HTM / HTML |
| DOI | 10.4064/cm6573-10-2015 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1502.00498v2.pdf |
| Alternate Webpage(s) | https://doi.org/10.4064/cm6573-10-2015 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
