Cyclotomic units in function fields

Let $k$ be a global function field over the finite field $\mathbb {F}_{q}$ with a fixed place $\infty$ of degree 1. Let $K$ be a cyclic extension of degree dividing $q-1$, in which $\infty$ is totally ramified. For a certain abelian extension $L$ of $k$ containing $K$, there are two notions of the group of cyclotomic units arising from sign normalized rank 1 Drinfeld modules on $k$ and on $K$. In this article we compare these two groups of cyclotomic units.