Generalized Leibniz's rule on a two step nilpotent Lie group
Let $G$ be a two-step nilpotent Lie group. For a function $f$ on $G$ and a multiindex $\alpha \in N^{\dim{G}}$, let $T^{\alpha}f(x)=x^{\alpha}f(x)$. We find and prove a formula for $T^{\alpha}(f*g)$ for any $\alpha \in N^{\dim{G}}$ and Schwartz functions $f,g$, where $*$ is the convolution on G. As an application we get the formula for $D^{\alpha}(f # g)$, where $f # g = (f^{\vee} * g^{\vee})^{\wedge}$ and $^{\wedge}$ is an Abelian Fourier Transform.