  For a $C^2$ function $u$ and an elliptic operator $L$, we prove a
quantitative estimate for the derivative $du$ in terms of local bounds on $u$
and $Lu$. An integral version of this estimate is then used to derive a
condition for the zero-mean value property of $\Delta u$. An extension to
differential forms is also given. Our approach is probabilistic and could
easily be adapted to other settings.
