Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have
$$\|f\|\_{L^\infty}\leq C\|f\|\_{L^p}^{1-\frac{1}{p}}\|f\|\_{W^{1,p}}^{\frac{1}{p}}?$$
My Motivation is the following: For $m \in \mathbb{N}$ I have an approximation operator $Q^h$ depending on the meshwidth $h>0$ such that for every $l\leq m$ there exist $C>0$ such that for every $f \in W^{m,p}$ we have
$$\|f-Q^hf\|\_{W^{l,p}}\leq Ch^{m-l}\|f\|\_{W^{m,p}}.$$
Now I want to show that $\|f-Q^hf\|\_{W^{m-1,\infty}} \rightarrow 0$. The statement above would imply that
$$\|f-Q^hf\|\_{W^{m-1,\infty}}\leq C h^{1-\frac{1}{p}}\|f\|\_{W^{m,p}}.$$



Estimate infinity norm with Lp and W1p norm
Yes. This is a known inequality, called e.g. "Gagliardo-Nirenberg inequality" in Brezis' book, Comment 1 to Chapter 8, page 233.


