Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?

Question

I have been reflecting on this question, and want to share my thinking thus far. I'd be grateful for the community's inputs.

We refer to Morrey's inequality, Theorem 4 on pp 266 of Evan's book on PDE, Vol 19, for a $C'$ bounded open set $U$ of $\mathbb{R}^n$, where the dimension $n \geq 2$ is 'apriori given' (that is, what one decides to work with) and a choice of the Lesbegue index $p$ appropriate to $n$. Thus, Morrey's inequality is oblivious to the important information of Nuclearity of $W^{(1,p)}_0 (U)$ and the equivalent norm on it shown in the related post:

Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$

Thus my 'apriori answer' is NO, which I wish to confirm.