Carleson measures on domains in Heisenberg groups

We study the Carleson measures on NTA and ADP domains in the Heisenberg groups $\mathbb{H}^n$ 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 Kor\'anyi--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 $\mathbb{H}^n$. Finally, we prove a Fatou-type theorem on $(\epsilon, \delta)$-domains in $\mathbb{H}^n$.