Local regularity for nonlocal double phase equations in the Heisenberg group

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group ${\mathbb{H}}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for nonlinear integro-differential problems in the Heisenberg group ${\mathbb{H}}^n$. Our proof of the a priori estiamtes bases on the spirit of De Giorgi-Nash-Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev-Poincaré type inequality in local domain, which may be of independent interest and potential applications.