{"title":"海森堡群域上的卡列森度量","authors":"Tomasz Adamowicz, Marcin Gryszówka","doi":"arxiv-2409.01096","DOIUrl":null,"url":null,"abstract":"We study the Carleson measures on NTA and ADP domains in the Heisenberg\ngroups $\\mathbb{H}^n$ and provide two characterizations of such measures: (1)\nin terms of the level sets of subelliptic harmonic functions and (2) via the\n$1$-quasiconformal family of mappings on the Kor\\'anyi--Reimann unit ball.\nMoreover, we establish the $L^2$-bounds for the square function $S_{\\alpha}$ of\na subelliptic harmonic function and the Carleson measure estimates for the BMO\nboundary data, both on NTA domains in $\\mathbb{H}^n$. Finally, we prove a\nFatou-type theorem on $(\\epsilon, \\delta)$-domains in $\\mathbb{H}^n$.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"41 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Carleson measures on domains in Heisenberg groups\",\"authors\":\"Tomasz Adamowicz, Marcin Gryszówka\",\"doi\":\"arxiv-2409.01096\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the Carleson measures on NTA and ADP domains in the Heisenberg\\ngroups $\\\\mathbb{H}^n$ and provide two characterizations of such measures: (1)\\nin terms of the level sets of subelliptic harmonic functions and (2) via the\\n$1$-quasiconformal family of mappings on the Kor\\\\'anyi--Reimann unit ball.\\nMoreover, we establish the $L^2$-bounds for the square function $S_{\\\\alpha}$ of\\na subelliptic harmonic function and the Carleson measure estimates for the BMO\\nboundary data, both on NTA domains in $\\\\mathbb{H}^n$. Finally, we prove a\\nFatou-type theorem on $(\\\\epsilon, \\\\delta)$-domains in $\\\\mathbb{H}^n$.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"41 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.01096\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01096","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 $\mathbb{H}^n$.