  In this paper, we get a Liouville type theorem for the special Lagrangian
equation with a certain 'convexity' condition, where Warren-Yuan first studied
the condition in [30]. Based on Warren-Yuan's work, our strategy is to show a
global Hessian estimate of solutions via the
Neumann-Poincar$\mathrm{\acute{e}}$ inequality on special Lagrangian graphs,
and mean value inequality for superharmonic functions on these graphs, where we
need geometric measure theory. Moreover, we derive interior Hessian estimates
on the gradient of the solutions to the equation with this 'convexity'
condition or with supercritical phase.
