Sobolev inequality and C^{1,1} estimate

Question

I read in an article the following statement:

I have question about how we get the inequality (2.19) :it is written that we uses $C^{1,1}$ estimate but here $p\geq 2 $ and not necesserly bigger than number $n$. So I can't use the Sobolev inequalities. so could you please give me an idea ?

Note that $B_{4}=\{x\in R^{n},|x|<4 \} $ and $\overline{\nabla u}=\int_{B_{4}}\frac{\nabla u}{|B_{4}|}$

Thanks for helping me