Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała. "Leibniz's rule on two-step nilpotent Lie groups." Colloquium Mathematicae 145.1 (2016): 137-148. <http://eudml.org/doc/286085>.

@article{KrystianBekała2016,
abstract = {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.},
author = {Krystian Bekała},
journal = {Colloquium Mathematicae},
keywords = {Leibniz rule; Heisenberg group; Fourier transform; homogeneous groups; symbolic calculus; convolution},
language = {eng},
number = {1},
pages = {137-148},
title = {Leibniz's rule on two-step nilpotent Lie groups},
url = {http://eudml.org/doc/286085},
volume = {145},
year = {2016},
}

TY - JOUR
AU - Krystian Bekała
TI - Leibniz's rule on two-step nilpotent Lie groups
JO - Colloquium Mathematicae
PY - 2016
VL - 145
IS - 1
SP - 137
EP - 148
AB - 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.
LA - eng
KW - Leibniz rule; Heisenberg group; Fourier transform; homogeneous groups; symbolic calculus; convolution
UR - http://eudml.org/doc/286085
ER -