  We introduce the notion of formally self-adjoint conformally covariant
polydifferential operators and give some constructions of families of such
operators. In one direction, we show that any homogeneous conformally
variational scalar Riemannian invariant (CVI) induces one of these operators.
In another direction, we use the ambient metric to give alternative
constructions of certain operators produced this way, one of which is a
formally self-adjoint, fourth-order, conformally covariant tridifferential
operator which should be regarded as the simplest fully nonlinear analogue of
the Paneitz operator.
