Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

Abstract: In this paper, we investigate the regularity for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Inspired by the De Giorgi-Nash-Moser theory, the local boundedness of weak subsolutions and the H\"{o}lder continuity of weak solutions to mixed local and nonlocal degenerate elliptic equations are established by deriving the Caccioppoli type inequality for weak subsolutions and the logarithmic estimates for weak supersolutions. Furthermore, the Harnack inequality for weak solutions and the weak Harnack inequality for weak supersolutions are proved by using the estimates involving a Tail term and expansion of positivity.