Let
- $d\in\mathbb N$
- $\Omega\subseteq\mathbb R^d$ be open
- $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $$\mathfrak D(\Omega):=\left\{\Phi\in\mathcal D(\Omega)^d:\nabla\cdot\Phi=0\right\}$$
In a paper, I've found the following theorem: Let $f\in\mathcal D'(\Omega)^d$ $\Rightarrow$ $$\left.f\right|_{\mathfrak D(\Omega)}=0\;\Leftrightarrow\;\exists p\in\mathcal D'(\Omega):f=\nabla p\;.\tag 1$$ This statement can be found in various books related to the study of the Navier-Stokes equations.
However, $(1)$ is obviously nonsense, cause by definition $f$ is a mapping $\mathcal D(\Omega)\to\mathbb R^d$ and the elements of $\mathfrak D(\Omega)$ are elements of $\mathcal D(\Omega)^d$, i.e. $\left.f\right|_{\mathfrak D(\Omega)}$ is an undefined symbol sequence.
So, at a first glance I thought they possibly mean $\left(\mathcal D(\Omega)^d\right)'$ instead of $\mathcal D'(\Omega)^d$. However, they write $f=(f_1,\ldots,f_d)$ in the paper (what suggests that $f$ is an $\mathbb R^d$-valued mapping) and by definition $\nabla p$ is a mapping $\mathcal D(\Omega)^d\to\mathbb R$.
So, my question is: How do I need to interpret this theorem?