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I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it. Let $u$ be a solution of $$\Delta u = f \;\;\; x \in B_4 $$ Then if we can bound $$\int_{B_4} |D^2 u|^2 \leq 1,\;\;\; \int_{B_4} |f|^2 \leq \delta^2$$ then $$\int_{B_4} |\nabla u - \overline{\nabla u}_{B_4}|^2 \leq C_1.$$ Let $v$ be the solution to $$\begin{cases} \Delta v = 0 & \\ v = u - (\overline{\nabla u})_{B_4}\cdot \vec{x} - \overline{u}_{B_4} & \partial B_4 \end{cases}.$$ Then by minimality of harmonic function with respect to energy in $B_4$, $$\int_{B_4} |\nabla v|^2 \leq \int_{B_4} |\nabla u - \overline{\nabla u}_{B_4}|^2 \leq C_1.$$

I understand how one uses minimality of the solution in the last part, but I'm not sure how you get the $C_1$ bound on $\nabla u - \overline{\nabla u}_{B_4}$. In particular, I'm not sure what the overline means. Restricted to the boundary?

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Here, the overline means averaging the function over the indicated domain, namely $B_4$. The Poincaré inequality allows us to estimate the $L^p$ norm of a function with mean zero in terms of the $L^p$ norm of its gradient. In particular, we can estimate the $L^2$ norm of $\nabla u-\overline{\nabla u}_{B_4}$ in terms of the $L^2$ norm of $D^2u$. From what you wrote I cannot tell what role the estimate with $\delta$ has in all of this. A reference to where you encountered these estimates might be helpful.

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