Carleson measures on domains in Heisenberg groups

We study the Carleson measures on nontangentially accessible (NTA) and admissible for the Dirichlet problem (ADP) domains in the Heisenberg groups Hn$\mathbb {H}^n$ 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 L2$L^2$‐bounds for the square function Sα$S_{\alpha }$ of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in Hn$\mathbb {H}^n$. Finally, we prove a Fatou‐type theorem on (ε,δ)$(\varepsilon, \delta)$‐domains in Hn$\mathbb {H}^n$. Our work generalizes results by Capogna–Garofalo and Jerison–Kenig.