On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields

Let $k$ be any number field and $k_{\infty}/k$ any $\mathbb{Z}_p$-extension. We construct a natural $\Lambda= \mathbb{Z}_p[[ T-1 ]]$-morphism from $\varprojlim k_n^{\times} \otimes_{\mathbb{Z}} \mathbb{Z}_p$ into a special subset of $C^1(\mathbb{Z}_p)^*$, the collection of linear functionals on the set of continuously differentiable functions from $\mathbb{Z}_p \to \mathbb{C}_p$. We apply the results to the problem of interpolating Gauss sums attached to Dirichlet characters and the explicit annihilation of real ideal classes.