My question is pretty straightfoward: I would like to be able to think of "the set of weakly differentiable functions". Recall that:
- a function $f:I=(a,b) \rightarrow \mathbb{R}$ is weakly differentiable, with derivative $f'$, if for every $\varphi \in C^\infty_c(I)$ it holds that $ \int_I f'\varphi dx = - \int_I f \varphi' dx $
This seems to imply that $f \in L^1_{loc}([a,b])$. However, a priori, it doesn't explicitly say something about the class of the derivative: is $f'\in L^1, L^2, L^\infty$..?
Is it correct to classify the set of weakly differentiable functions on $[a,b]$ as $W^{1,1}([a,b])$ (or maybe better $W^{1,1}_{loc}([a,b])$)?
If not, is there a notation for such a set?