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} α$\mathcal{M}$^+{λ,Λ}\big(D^2{\mathbbm{H}^N,S}u\big)-β\big(-Δ_{\mathbbm{H}^N}\big)^su=f &$\text{in }$ ,Ω, u=g &$\text{in }$ ,\mathbbm{H}^N\setminusΩ, \end{cases} \end{align} where ${\mathcal{M}}_{\lambda ,\mathrm{\Lambda }}^{+}$ is the extremal Pucci’s operator and $(-{\mathrm{\Delta }}_{\text{mathbbm}{H}^N}{)}^s$ denotes the fractional sub-Laplacian on Heisenberg group. Here $\alpha \ge 0$ and $\beta >0$ are constants.