Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|\_{L^q(I)} + \sum\_{j = 0}^{m - 2} \|D^ju\|\_{L^\infty(I)} \le \epsilon\|D^mu\|\_{L^1(I)} + C\|u\|\_{L^1(I)}$$for all $u \in W^{m, 1}(I)$?



Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant
We argue by contradiction. After scaling, we have a sequence $(u\_n)\_{n=1}^\infty \subset W^{m,1}(I)$ with
$$
\| D^{(m-1)} u\_n \|\_q + \sum\_{j=0}^{m-2} \| D^j u\_n \|\_\infty = 1 > \varepsilon \| D^m u\_n \|\_1 + n \| u\_n \|\_1.
$$
We have $(D^{m-1}u\_n)$ is a bounded subset of $W^{1,1}(I)$ from
$$
\| D^{m-1} u\_n \|\_1 \leq \| D^{m-1} u\_n \|\_q \leq 1, \quad
\| D^{m} u\_n \|\_1 < \varepsilon^{-1}.
$$
 Furthermore, $(D^j u\_n)$ is a bounded subsequence of $W^{1,1}(I)$ for $0 \leq j \leq m-2$ from
 $$
 \| D^{j} u\_n \|\_1 \leq \| D^{j} u\_n \|\_\infty \leq 1, \quad \| D^{m-1} u\_n \|\_1 \leq \| D^{m-1} u\_n \|\_q \leq 1.
 $$
 By the compact embedding $W^{1,1}(I) \subset L^q(I)$, we may assume, after passing to subsequences, that $D^j u\_n \to g\_j$ in $L^q(I)$ for $0 \leq j \leq m-1$, for some $g\_j \in L^q(I)$. It is clear that $u\_n \to 0$ in $L^1(I)$. Thus
 $$
 \| u\_n \|\_q \leq \| u\_n \|\_\infty^{(q-1)/q} \| u\_n \|\_1^{1/q} \leq \| u\_n \|\_1^{1/q} \to 0.
 $$
 Thus $(u\_n)$ converges to $0$ in $W^{m-1,q}(I)$. Since $I$ is finite, there is some $C$ for which
 $$
 \| D^j u\_n \|\_\infty \leq C \| D^j u\_n \|\_{W^{1,q}} \quad \forall 0 \leq j \leq m-2.
 $$
 Thus $\| D^j u\_n \|\_\infty \to 0$ for all $0 \leq j \leq m-2$. We have also shown $\| D^{(m-1)} u\_n \|\_q \to 0$, but by assumption
 $$
\| D^{(m-1)} u\_n \|\_q + \sum\_{j=0}^{m-2} \| D^j u\_n \|\_\infty = 1,
 $$
 contradiction.


