We establish the comparison principle, existence and regularity of solutions to the following problem concerning the mixed operator: \(\begin{aligned} {\left\{ \begin{array}{ll} \alpha \mathcal {M}^+_{\lambda ,\Lambda }\big (D^2_{\mathbbm {H}^N,S}u\big )-\beta (-\Delta _{\mathbbm {H}^N})^su=f & \text {in } \,{\Omega },\\ u=g & \text {in } \,\mathbbm {H}^N\setminus \Omega , \end{array}\right. } \end{aligned}\) where \(\mathcal {M}_{\lambda ,\Lambda }^+\) is the Pucci’s extremal operator and \((-\Delta _{\mathbbm {H}^N})^s\) denotes the fractional sub-Laplacian on the Heisenberg group \(\mathbbm {H}^N.\)
