By Caldrron-Zygmund inequality, you may obtain :
 $$  \|u\|_{W^{2,1+\delta}~~(B_{\frac{1}{4}})} \leq C\|u\|_{L^{1+\delta}~~(B_{\frac{1}{2}})} +C\|g\|_{L^{1+\delta}~~(B_{\frac{1}{2}})}.$$
Since the dimension of space is 2, 
$$ \|u\|_{L^{\infty}~~(B_{\frac{1}{4}})}\leq  C\|u\|_{W^{2,1+\delta}~~(B_{\frac{1}{4}})}.$$
Using the $L^1$- theory, see
[![enter image description here][1]][1]

you can obtain 
$\|u_1\|_{L^{1+\delta} ~~(B_{\frac{1}{2}})}$ is bounded. By the mean value theorem or harnack inequality you can obtain $\|u_3\|_{L^{\infty}~~(B_{\frac{1}{2}})}\leq C$, combine with the fact that $\|u_2\|_{L^{\infty}~~(B_{\frac{1}{2}})}\leq C$ you may obtain $$\|u\|_{L^{1+\delta}~~(B_{\frac{1}{2}})} \leq C,$$
and finish the proof.


  [1]: https://i.sstatic.net/gC1RQ.png