  We present a conformal deformation involving a fully nonlinear equation in
dimension 4, starting with positive scalar curvature. Assuming a certain
conformal invariant is positive, one may deform from positive scalar curvature
to a stronger condition involving the Ricci tensor. We also give a new
conformally invariant condition for positivity of the Paneitz operator, which
allows us to give many new examples of manifolds admitting metrics with
constant $Q$-curvature.
