ORIGINAL ARTICLE
Carleson measures on domains in Heisenberg groups
Tomasz Adamowicz,
Marcin Gryszówka,
Corresponding Author
Tomasz Adamowicz
The Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Correspondence
Tomasz Adamowicz, The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland.
Email: [email protected]
Search for more papers by this authorMarcin Gryszówka
The Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Search for more papers by this authorTomasz Adamowicz,
Marcin Gryszówka,
Corresponding Author
Tomasz Adamowicz
The Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Correspondence
Tomasz Adamowicz, The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland.
Email: [email protected]
Search for more papers by this authorMarcin Gryszówka
The Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Search for more papers by this authorFirst published: 22 June 2025
T. A. and M. G. were supported by the National Science Center, Poland (NCN), UMO-2020/39/O/ST1/00058.
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.
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