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 $H^n$ 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 $L^2$-bounds for the square function $S_{\alpha }$ of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in $H^n$. Finally, we prove a Fatou-type theorem on $(\epsilon ,\delta )$-domains in $H^n$. Our work generalizes results by Capogna–Garofalo and Jerison–Kenig.