I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.

In particular, let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary, and define $W^{k,p,j}(\Omega, \mathbb{R}^m)$ to be the Sobolev space of $L^p$ functions $\Omega \rightarrow \mathbb{R}^m$ with norm $$ \|f\|_{W^{k,p,j}(\Omega, \mathbb{R}^m)} = \sum_{\alpha \le k} \left[ \int_{\Omega} \|\partial^{\alpha} f(x)\|_j^p \, dx \right]^{1/p} $$ where $\partial^{\alpha}$ indicates the weak derivative with multi-index $\alpha$.

I'm likely okay if any additional simplifying assumptions are needed. Notably, I do not need any results for the general Banach-space-valued case, which I assume could get quite technical. I saw that there is some work on $k = 1$ (see this MO question and also this this master thesis).

infinite-dimensionalvector space. That is much more general than what you seem to need. $\endgroup$ – user101142 Aug 5 '18 at 19:04