Take a look at the inhomogeneous Laplace equation on a ball $B$:
\begin{cases} - \Delta f = c\varphi, &\text{in } B \\ f = 0, & \text{on } \partial B \end{cases}
where $\varphi$ is Lipschitz and $c > 0$. (This is linked to another question of mine.)
Now, how do I get the estimate:
$$\sup_{B_{1/2}} f \leq \bigg( \int_B (c\varphi)^p \bigg)^\frac 1p$$
As always, thanks for any help!