The characterization of norm compactness in $L^p$: you want the [Frechét-Kolmogorov theorem][1], usually stated for $X=\mathbb{R}^n$ and $Y:=\mathbb{R}$. The proof follows quite easily form Ascoli-Arzelà by mollification. You can find it on  Haïm Brezis'  *Functional Analysis*.

[1]:https://en.wikipedia.org/wiki/Fréchet–Kolmogorov_theorem