I am reading Struwe's "Variational methods" and he sometimes uses the space $D^{k,p}(\Omega)$, defined as the closure of $C^{\infty}_0 (\Omega)$ with respect to the norm given by:
$$||u||_{D^{k,p}}^p= \sum_{|\alpha|=k} ||D^{\alpha} u||_p^p $$
The problem is that if $\Omega=\mathbb{R}^n$, for example, you don't have Poincaré inequality and thus that should be a different space than $W^{k,p}(\Omega)$ (and he uses a different name, indeed). The thing is that he uses embedding results known for Sobolev spaces with functions in $D^{k,p}$. If for instance you look at page 40 he says "By Sobolev's embedding $D^{k,p} \hookrightarrow L^q$ with $\frac{1}{q}=\frac{1}{p} - \frac{k}{n}$".
So, is there a simple reason for this embedding to hold? Also, do these spaces have a particular name? I couldn't find anything
If $\Omega$ is bounded, then $D^{k,p}=W^{k,p}_0$.
The Sobolev embedding theorem for $D^{k,p}$ can be deduced the following way. If you look into standard proofs for embedding of $W^{k,p}_0$ then the result is proven for functions from $C_c^\infty$ first. Interestingly, lower order Sobolev norms are estimate against only $D^{k,p}$-(semi)norm of the smooth function. See, e.g., Evans: Theorem 1 in 5.6.1. Hence, these embeddings carry over by density to the space $D^{k,p}$.