Here, $\Omega\subseteq\mathbb{R}^n$ is an open set, $X$ is a Banach space, and $X^*$ its topological dual. I ended up on a problem for which I need to define a suitable notion for the gradient of a map $\mu:\Omega\to X$, and I would need this object to act as linear continuous maps on smooth functions $\phi:\Omega\to (X^*)^n$, i.e.:
$$ \nabla\mu \in C^\infty_c(\Omega;(X^{*})^n)^* $$
In this context, I would need to see both $\mu$ and $\nabla\mu$ as vector-valued distributions, in some adequate sense.
So far, I have only found one reference, namely Bhattacharyya - Distributions. Generalized functions with applications in Sobolev spaces. Also, this question seems to be related, but it doesn't seem to answer my doubts.
However, I have two issues.
First, Bhattacharyya always considers a one-dimensional domain $I\subseteq\mathbb{R}$, instead of an $\Omega\subseteq\mathbb{R} ^n$. However, maybe this can be easily overcome by defining "$\nabla = (\partial_1,\dots, \partial_n)$", and working on partial derivatives.
Second, and most importantly, the theory is developed for distributions
$$ \mathcal{L}_c(C^\infty_c(I);X) $$
i.e. distributions are linear continuous objects mapping $C^\infty_c(I)\to X$, rather than linear continuous objects mapping $C^\infty_c(I;X)\to \mathbb{R}$.
Is anyone aware of a reference that is more suited to my setting?
EDIT: thanks to this question I discovered Amann's work on vector-valued distributions. One result in my direction, in Amann - Linear and Quasilinear Parabolic Problems (2019) , Theorem VI.1.3.1, states that (if I am not missing something) $$ S^*(\mathbb{R}^n;X^*) = S(\mathbb{R}^n;X)^*, $$ where $S(\mathbb{R}^n;X)$ is the space of Banach-valued Schwartz functions, $S^*(\mathbb{R}^n;X^*) = \mathcal{L}_c(S(\mathbb{R}^n);X^*))$ and $S(\mathbb{R}^n;X)^*$ just stands for the topological dual.
I even find it hard to discern the isomorphism map (I think it must come from some density of $S(\mathbb{R}^n)\times X$ in $S(\mathbb{R};X)$ as well as of $S^*(\mathbb{R}^n)\times X$ in $S^*(\mathbb{R};X)$). Also, that equivalence is only stated for Schwartz distributions, it doesn't mention if it holds for standard distributions (i.e. $C^\infty_c$) too...
