  For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero
boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq
(2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2
\,dx\right)^{1/4}. \] The method of proof is elementary and admits
generalizations. The inequality is applied to establish an existence theorem
for the Burgers equation.
