Definition 5

Let $G$ be a Lie group and $t \in U(G)$. Let $L_t$ denote the field of distributions $g \mapsto \varepsilon_g * t$ on $G$ and $R_t$ the field of distributions $g \mapsto t * \varepsilon_g$ on $G$.

In other words, $L_t$ (resp. $R_t$) is the field of distributions defined by $t$ and $G$ operating on $G$ on the right (resp. left) by means of the mapping $(g, g') \mapsto gg'$. Let $\Omega$ be an open subset of $G$ and $F$ a Hausdorff polynormed space; if $f \in C^\omega(\Omega, F)$, then $L_t f = f * t^\vee \in C^\omega(\Omega, F)$ and

(no. 5). If $G$ is finite-dimensional, the differential operators $L_t$ and $R_t$ are of class $C^\omega$ (no. 5).