  We introduce the notion of $ P -$functions for fully nonlinear equations and
establish a general criterion for obtaining such quantities for this class of
equations. Some applications are gradient bounds, De Giorgi-type properties of
entire solutions and rigidity results. Particularly, we establish a gradient
bound and a rigidity result for Pucci's equations. Furthermore, we prove
Harnack-type inequalities and local pointwise estimates for the gradient of
solutions to fully nonlinear elliptic equations. In addition, we consider such
quantities for higher order nonlinear equations and for equations of order
greater than two we obtain Liouville-type theorems and pointwise estimates for
the Laplacian.
