Carleson measures on domains in Heisenberg groups

TL;DR

This paper characterizes Carleson measures on NTA and ADP domains in Heisenberg groups using harmonic functions and quasiconformal mappings, establishing bounds and boundary behavior theorems.

Contribution

It provides new characterizations of Carleson measures in Heisenberg groups and proves boundary limit theorems for harmonic functions on these domains.

Findings

Characterization of Carleson measures via level sets of harmonic functions.

Establishment of $L^2$ bounds for the square function $S_{\alpha}$.

Proof of a Fatou-type boundary theorem for $(\epsilon, \delta)$-domains.

Abstract

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 $\mathbb{H}^n$.