I've been reading through the Sobolev spaces chapter of Lawrence Evan's book on PDE's and am struggling to understanding one of the steps in their proof.
I've reached the point where we have the inequality $$\lVert u_m^\epsilon-u_m \rVert_{L^q(V)} \leq \lVert u_m^\epsilon-u_m \rVert_{L^1(V)}^\theta \lVert u_m^\epsilon-u_m \rVert_{L^{p*}(V)}^{1-\theta}$$ where $u_m^\epsilon$ is the mollification of $u_m$. We showed earlier that we can make the $L^1$ term as small as we wish, which means that if we can bound the $p*$ term, we will have similar control on the $q$ norm of this difference.
The way he suggests of doing this is first by appealing to Sobolev-Gagliardo-Nirenberg, which would give $$\lVert u_m^\epsilon-u_m \rVert_{L^{p*}(V)}\leq \lVert u_m^\epsilon-u_m \rVert_{W^{1,p}(V)}$$ and then notes that, because we have a bound for $$\sup_{m\in\mathbb{N}}\lVert u_m \rVert_{W^{1,p}(V)}$$ we can bound the $p*$ norm term above uniformly in $m$.
I feel like it shouldn't be too difficult to find a way to apply this bound to the difference above, but I cannot find any way to do it. The only thing I can think of is to try to make an appeal to one of the approximation theorems, but I don't think this makes any use of the bound we have for the Sobolev norms of the $u_m$.
First, recall Young’s inequality for convolutions: if $f\in L^1(\Bbb{R}^n)$ and $g\in L^p(\Bbb{R}^n)$, then $f*g\in L^p(\Bbb{R}^n)$ and $\|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}$. In particular, if $g$ is supported inside a set $V$, then \begin{align} \|f*g\|_{L^p(V)}\leq \|f*g\|_{L^p(\Bbb{R}^n)}\leq \|f\|_{L^1(\Bbb{R}^n)}\|g\|_{L^p(\Bbb{R}^n)}=\|f\|_{L^1(\Bbb{R}^n)}\|g\|_{L^p(V)}. \end{align}
Now, if $g\in W^{k,p}(\Bbb{R}^n)$ has compact support inside an open set $V$, then its regularization $g^{\epsilon}=\eta_{\epsilon}*g$ is such that \begin{align} \|g^{\epsilon}\|_{W^{k,p}(V)}=\sum_{|\alpha|\leq k}\|D^{\alpha}(\eta_{\epsilon}*g)\|_{L^p(V)}=\sum_{|\alpha|\leq k}\|\eta_{\epsilon}*D^{\alpha}g\|_{L^p(V)} &\leq\sum_{|\alpha\leq k}\|\eta_{\epsilon}\|_{L^1(\Bbb{R}^n)}\|D^{\alpha}g\|_{L^p(V)}\\ &=\|\eta\|_{L^1(\Bbb{R}^n)}\|g\|_{W^{k,p}}(V), \end{align} where I used the fact that by a simple change of variables, all the $\eta_{\epsilon}$ have the same $L^1$ norm.
So, for the actual question, by the triangle inequality, we have that \begin{align} \|u_m^{\epsilon}-u_m\|_{W^{1,p}(V)}\leq (\|\eta\|_{L^1(\Bbb{R}^n)}+1)\left(\sup\limits_{j\in\Bbb{N}}\|u_j\|_{W^{1,p}(V)}\right). \end{align} So, the quantity on the left is uniformly bounded in $m$ and $\epsilon$.