Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|\_{L^p(0, 1)} \le \epsilon \|u'\|\_{L^1(0, 1)} + C\|u\|\_{L^1(0, 1)} \text{ for all }u \in W^{1, 1}(0, 1)?$$



Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$?
Of course. 


For $u\in C^1$ denote $v=u-\int\_0^1 u$, then $v(x\_0)=0$ for some $x\_0\in (0,1)$ and for all $x\in (0,1)$ we have $$|v(x)|=|v(x)-v(x\_0)|=\left|\int\_{x\_0}^x v'(t)dt\right|\leqslant \|v'\|\_{L^1(0,1)}=\|u'\|\_{L^1(0,1)},$$
thus $M:=\|v\|\_{L^{\infty}(0,1)}\leqslant \|u'\|\_{L^1(0,1)}$. This extends to $u\in W^{1,1}$ by continuity. We have $\|u\|\_p\leqslant \|v\|\_p+\|u\|\_1$. Next,
$$
\|v\|\_p^p=\int |v(x)|^p dx\leqslant M^{p-1} \int |v(x)| dx= 
(\varepsilon^{-1} \|v\|\_1^p)^{1/p}\cdot (M^{p}
\varepsilon^{1/(p-1)})^{1-1/p}\leqslant \varepsilon^{-1} \|v\|\_1^p+ M^{p}
\varepsilon^{1/(p-1)},
$$
and result follows from this inequality and $\|v\|\_1\leqslant 2\|u\|\_1$.


