Two versions of the Nikodym maximal function on the Heisenberg group
type: Item
instance of: scholarly article
title: Two versions of the Nikodym maximal function on the Heisenberg group
zbMATH Open document ID: 1177.43007
author: Joonil Kim
published in: Journal of Functional Analysis
publication date: 2009-09-10T00:00:00Z
review text: The Nikodym maximal function on the Euclidean plane \(\mathbb R^2\) is defined as the supremum over averages over rectangles of eccentricity \(N\), and its operator norm in \(L^2(\mathbb R^2)\) is known to be \(O(\log N)\). The author considers two variants, one on the standard Heisenberg group \(\mathbb H^1\) and the other on the polarized Heisenberg group \(\mathbb H_p^1\). The group law of \(\mathbb H_p^1\) is given by \(x\cdot_p y=(x_1+y_1,x_2+y_2, x_3+y_3+x_1y_2)\). Associated with the two cases of Heisenberg groups, the author considers two maximal averages over all rectangles \(R\) of eccentricity \(N\) supported on the hyperplane \(\Pi=\{(x_1,x_2,0); x_1,x_2\in\mathbb R\}\): on \(\mathbb H^1\), \(\mathcal M_Nf(x)=\sup_{R}|R|^{-1}\int_R|f(x\cdot(y_1,y_2,0))|dy_1dy_2\), and on \(\mathbb H_p^1\), \(\mathcal M_N^pf(x)=\sup_{R}|R|^{-1}\int_R|f(x\cdot_p(y_1,y_2,0))| dy_1dy_2\). His main results are: (1) There are \(c,C>0\) such that \(c\log N\leq \|\mathcal M_N^p\|_{L^2(\mathbb H_p^1)\to L^2(\mathbb H_p^1)} \leq C(\log N)^{3/2}\). (2) There are \(c,C>0\) such that \(cN^{1/4}\leq \|\mathcal M_N\|_{L^2(\mathbb H^1)\to L^2(\mathbb H^1)} \leq CN^{1/4}(\log N)^{2}\).
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47A80
zbMATH DE Number: 5602664
zbMATH Keywords: maximal operators
zbMATH Keywords: Nikodym maximal function
zbMATH Keywords: Heisenberg group
zbMATH Keywords: oscillatory integral operator
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.05.020
OpenAlex ID: W2002393563
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DOI: 10.1016/J.JFA.2009.05.020
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description: scientific article; zbMATH DE number 5602664