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The nonnegativity of the CR Paneitz operator plays a crucial role inthree-dimensional CR geometry. In this paper, we prove this nonnegativity forembeddable CR manifolds. This result and previous works give an affirmativesolution of the CR Yamabe problem for embeddable CR manifolds. We also show theexistence of a contact form with zero CR $Q$-curvature, and generalize thetotal $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einsteincontact forms. Furthermore, we discuss the logarithmic singularity of theSzeg\H{o} kernel.

Nonnegativity of the CR Paneitz operator for embeddable CR manifolds