Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group

We establish the comparison principle, existence and regularity of viscosity solutions to the following problem concerning the mixed operator: \begin{align} \begin{cases} \alpha\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\beta\big(-\Delta_{\mathbbm{H}^N}\big)^su=f &\text{in } \,{\Omega}, u=g &\text{in } \,\mathbbm{H}^N\setminus\Omega, \end{cases} \end{align} where $\mathcal{M}_{\lambda,\Lambda}^+$ is the extremal Pucci's operator and $(-\Delta_{\mathbbm{H}^N})^s$ denotes the fractional sub-Laplacian on Heisenberg group. Here $\alpha\geq 0$ and $\beta>0$ are constants.