How can I show that the unity ball $B_1:=\{f\in W^{1,p}([a,b]) \text{ : } ||f||_{1,p}=||f||_{L^p}+||f'||_{L^p}\leq1\} $ is precompact in $L^p([a,b])$?
I thought of using the Riesz-Kolmogorov theorem for this, which states that a subset $M\subseteq L^p([a,b]), a\leq p<\infty, -\infty<a<b<\infty$, is precompact iff
- $M$ is bounded and
- $\lim \limits_{h\to0^+}\sup\limits_{f\in M}\int_a^{b-h}|f(t+h)-f(t)|^pdt=0$.
I know that $B_1$ is bounded using the fact that all $f\in B_1$ also belong to $W^{1,p}$, i.e $|f(x)|\leq |f(a)|+\int_a^x|g(t)|dt \in \mathbb{R}$, where $g\in L^1([a,b])$ is the weak derivative of $f$, but I have trouble proving the second condition. I know that translations are continuous in $L^p$, and the fact that $||f||_{1,p}\leq1$ provides that $f,f'\in L^p([a,b])$, but I don't know how to deal with the sup.
Any suggestions ?