Carleson measures on domains in Heisenberg groups

Abstract We study the Carleson measures on nontangentially accessible (NTA) and admissible for the Dirichlet problem (ADP) domains in the Heisenberg groups and provide two characterizations of such measures: (1) in terms of the level sets of subelliptic harmonic functions and (2) via the 1‐quasiconformal family of mappings on the Korányi–Reimann unit ball. Moreover, we establish the ‐bounds for the square function of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in . Finally, we prove a Fatou‐type theorem on ‐domains in . Our work generalizes results by Capogna–Garofalo and Jerison–Kenig.