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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).

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    $\begingroup$ Can't you recover the embeddings for $\mathbb{R}^m$-valued Sobolev maps from the corresponding componentwise embeddings? $\endgroup$
    – Teri
    Commented Aug 5, 2018 at 18:29
  • $\begingroup$ Both of your links are concerned with functions taking values in an infinite-dimensional vector space. That is much more general than what you seem to need. $\endgroup$
    – user101142
    Commented Aug 5, 2018 at 19:04
  • $\begingroup$ @Teri Hm, maybe it is just that simple. I guess then this is just an exercise. I guess then my only question would be whether that is sharp. But maybe that is also an exercise. $\endgroup$
    – Christopher A. Wong
    Commented Aug 5, 2018 at 21:51
  • $\begingroup$ What is $\Vert\cdot\Vert_j^p$? What does $j$ stand for? $\endgroup$
    – Piotr Hajlasz
    Commented Jan 5, 2019 at 20:13

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