  We consider here operators which are sum of (possibly) fractional
derivatives, with (possibly different) order. The main constructive assumption
is that the operator is of order~$2$ in one variable. By constructing an
explicit barrier, we prove a Lipschitz estimate which controls the oscillation
of the solutions in such direction with respect to the oscillation of the
nonlinearity in the same direction.
  As a consequence, we obtain a rigidity result that, roughly speaking, states
that if the nonlinearity is independent of a coordinate direction, then so is
any global solution (provided that the solution does not grow too much at
infinity). A Liouville type result then follows as a byproduct.

Analysis of PDEs