I stumpled over the following result in a script:
Let $1 \leq p < \infty$ and $f \in L_p[a,b]$. Define the Volterra operator as $$Vf(t) = \int_a^t f(s) ds.$$ Then we have $Vf \in W^{1, p}[a,b]$ and $(Vf)' = f$.
Now there is given a hint: It is sufficient to consider the case $p = 1$ and $f \in C^1[a,b]$.
I could prove the statement for $p = 1$ and $f \in C^1[a,b]$ but I don't quite get why that should be sufficient.
I would be grateful for your advice :)
You can approximate any function in the Lebesgue space arbitrarily well by a differentiable function.
Taking a converging sequence of such approximating functions you can show that $ Vf_n $ is Cauchy in the Sobolev space, hence has a limit in the Sobolev space. Since $ V $ is continuous as operator into the Lebesgue space the Sobolev limit equals $ Vf $