pertaining to the CR Paneitz operator. See Theorem \[t2\] below for more details.

As for the proof of the case of vanishing pseudohermitian torsion :

(Proof of Corollary \[c2\])

Setting $A_{11}=0$ in $\left(  \ref{13}\right)  $, by Theorem \[t1\], it suffices to show that $$\gamma_{1,0}=0$$ in order to have a globally defined pseudo-Einstein contact form $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$.

Note that, from $\left(  \ref{0}\right)  $ and $A_{11}=0$, $$R_{,1}=2u_{\overline{1}11}-i\gamma_{1,0}.\label{8}$$

Utilizing integration by parts, it follows from (\[8\]) and $\gamma
_{\overline{1},1}=0$ that $$\begin{array}
[c]{ccl}0 & \leq & \int_{M}\left \vert \gamma_{1,0}\right \vert ^{2}d\mu \\
& = & -\int_{M}\gamma_{1}\gamma_{\overline{1},00}d\mu \\
& = & -\int_{M}\gamma_{1}\left(  -iR_{,\overline{1}}+2iu_{,1\overline
{1}\overline{1}}\right)  _{,0}d\mu \  \\
& = & i\int_{M}\gamma_{1}\left(  R_{,0}-2u_{,1\overline{1}0}\right)
_{,\overline{1}}d\mu \\
& = & -i\int_{M}\gamma_{1,\overline{1}}\left(  R_{,0}-2u_{,1\overline{1}0}\right)  d\mu \\
& = & 0.
\end{array}
.$$

The third equality comes from $\left(  \ref{6}\right)  $ and $A_{11}=0$. Then $$\gamma_{1,0}=0.$$

Before giving the proof of Theorem \[t2\], we need the following Bôchner-type equality.

\[l8\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold and $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form. Then we have$${\displaystyle \int \limits_{M}}
\left(  2R-Tor\right)  \left(  \gamma,\gamma \right)  d\mu+2{\displaystyle \int \limits_{M}}
\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\frac{1}{2}{\displaystyle \int \limits_{M}}
\left(  P_{0}f\right)  fd\mu=0.\label{26A}$$

From Theorem \[t1\] and the commutation formula, it follows that$$\widetilde{\theta}=e^{\frac{\left(  