pseudoconvex CR $3$-manifolds.

\[t2\] If $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$ and nonnegative CR Paneitz operator $P_{0} $. Assume that the pseudohermitian curvature is $\left(  \frac{1}{2},0\right)
$-positive $$R>|A_{11}|.$$ Then $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f$ if and only if the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]).

We observe that, for a strictly pseudoconvex $3$-manifold $(M^{3},J,\theta)$, we have the invariance property for the CR-pluriharmonic operator $P_{1}$ and CR Paneitz operator $P_{0}$. It is to say that, for rescaled contact form $\widetilde{\theta}=e^{2g}\theta,$ we have $$\widetilde{P}_{1}=e^{-3g}P_{1}\text{ \textrm{and}\  \ }\widetilde{P}_{0}=e^{-4g}P_{0}.\label{A}$$ Then the nonnegativity of CR Paneitz operator $P_{0}$ is CR conformal invariant ([@h]).

Since the CR Paneitz operator $P_{0}$ is nonnegative ([@ccc]) if the pseudohermitian torsion is vanishing, it follows from Theorem \[t2\] and Corollary \[c2\] that

\[c3\] If $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$. Assume that the manifold is Sasakian and the Tanaka-Webster scalar curvature is positive, then we have the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]). That is to say that the Kohn–Rossi cohomology class of $\sigma_{\overline{1}}\theta^{\overline{1}}$ is vanishing.

Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0.$ With the notations 