  For the solution of the Poisson problem with an $L^\infty$ right hand side
\begin{equation*} \begin{cases} -\Delta u(x) = f (x) & \mbox{in } D,
  u=0 & \mbox{on } \partial D, \end{cases} \end{equation*} we derive an optimal
estimate of the form $$ \|u\|_\infty\leq \|f\|_\infty
\sigma_D(\|f\|_1/\|f\|_\infty), $$ where $\sigma_D$ is a modulus of continuity
defined in the interval $[0, |D|]$ and depends only on the domain $D$. In the
case when $f\geq 0$ in $D$ the inequality is optimal for any domain and for any
values of $\|f\|_1$ and $\|f\|_\infty.$ We also show that $$
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|], $$ where $B$ is a ball and
$|B|=|D|$. Using this optimality property of $\sigma,$ we derive
Brezis-Galloute-Wainger type inequalities on the $L^\infty$ norm of $u$ in
terms of the $L^1$ and $L^\infty$ norms of $f.$ The estimates have explicit
coefficients depending on the space dimension $n$ and turn to equality for a
specific choice of $u$ when the domain $D$ is a ball. As an application we
derive $L^\infty-L^1$ estimates on the $k-$th Laplace eigenfunction of the
domain $D.$
