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I've since come across the article Amann, Herbert. Compact embeddings of vector-valued Sobolev and Besov spaces. Dedicated to the memory of Branko Najman. Glas. Mat. Ser. III 35(55) (2000), no. 1, …

The following is stated in the paper Moshe Marcus and Victor J. Mizel, MR 531975 Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc. 251 (1 …

The following is known: Let $s \in (0,1)$ and $p \in [1,\infty)$ be such that $sp < n$. Let $q \in [1, p^*_{n,s})$ with $p^*_{n,s} = np/(n-sp)$, $\Omega \subset \mathbb R^n$ be a bounded extension …

The paper (and this comment on it) Peetre, Jaak. A remark on Sobolev spaces. The case $0<p<1$. Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, III. J …

Yes, this is correct; in fact, you could replace the map $|\cdot|$ with an arbitrary (uniformly) Lipschitz map $f$ (with $f(0) = 0$ if $G$ is unbounded) [1]. [1] Marcus, Moshe; Mizel, Victor J. Every …

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the Sobolev-to- …