Nonnegativity of the CR Paneitz operator for embeddable CR manifolds

The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a contact form with zero CR $Q$-curvature, and generalize the total $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einstein contact forms. Furthermore, we discuss the logarithmic singularity of the Szeg\H{o} kernel.