Conformal boundary operators associated to the Paneitz operator have attracted a lot of attention. We focus on two related topics: One is to establish a sharp Sobolev trace inequality on a three-ball, which is equivalent to a sharp Sobolev inequality on a two-sphere involving the fractional GJMS operator $P_3$; the other is to classify nonnegative solutions of a biharmonic equation on the half-space and unit ball with proper pair of conformal boundary operators, where the notions of biharmonic Poisson kernel and Green function are introduced. This is jointly with Shihong Zhang.