Extension of weak derivatives in Bochner spaces

Question

I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans:

He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In fact, the theory developed in the earlier section of difference quotients is for real-valued functions, not for functions with values in Banach spaces. What theorem did he use in fact? Any ideas?

Answer 1

The method of difference quotients is also true for functions with values in $L^2$. It can be found for instance in:

J.L. Lions. Equations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.