respect to $(J,\theta)$ essentially positive if there exists a constant $\Lambda$ $>$ $0$ such that $$\int_{M}P_{0}\varphi \cdot \varphi d\mu \geq \Lambda \int_{M}\varphi^{2}d\mu.\label{41}$$ for all real smooth functions $\varphi$ $\in(\ker P_{0})^{\perp}$ (i.e. perpendicular to the kernel of $P_{0}$ in the $L^{2}$ norm with respect to the volume form $d\mu$ $=$ $\theta \wedge d\theta).$ We say that $P_{0}$ is nonnegative if $$\int_{M}P_{0}\varphi \cdot \varphi d\mu \geq0$$ for all real smooth functions $\varphi$.

\[r1\] 1. The notions of Paneitz operator $P_{0}$ and $Q$-curvature were initially introduced on a Riemannian manifold, and were considered as a kind of generalization of Laplacian and Gaussian curvature on a two-dimensional manifold, respectively ([@h]).

2\. The kernel of the CR Paneitz operator $P_{0}$ is infinite dimensional, containing all CR-pluriharmonic functions.

3\. Let $(M,J,\theta)$ be a closed strictly pseudoconvex $3$-manifold with vanishing pseudohermitian torsion. Then the corresponding CR Paneitz operator $P_{0}$ is essentially positive ( [@ccc], [@cac]).

Finally, we define the CR $Q$-curvature in a pseudohermitian $3$-manifold by $$Q:=-\mathrm{Re}(R_{_{,}1}-iA_{11,\bar{1}})_{\bar{1}}=-\mathrm{Re}(R_{,1\bar
{1}}-iA_{11,\bar{1}\bar{1}}).\label{c}$$ Then $$Q=-\frac{1}{2}[\Delta_{b}R-i(A_{11,\bar{1}\bar{1}}-A_{\bar{1}\bar{1},11})].$$ Now for $\theta=e^{2\gamma}\theta_{0}$, under this conformal change, it is known that we have the following transformation laws ([@h]) : $$Q=e^{-4\gamma}(Q_{0}+\frac{3}{4}P_{0}\gamma)\label{0b}$$ and $$W_{1}:=(R_{,1}-iA_{11},_{\bar{1}})=e^{-3\gamma}[\overset{0}{R}_{,1}-i\overset{0}{A}_{11,\bar{1}}-6\overset{0}{P}_{1}\gamma],\label{0c}$$ where $P_{0}$ and $Q_{0}$ denote the CR Paneitz operator and the CR $Q$-curvature with respect 