  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.

Representation Theory;Functional Analysis