Lacunary directional maximal function on the Heisenberg group
type: Item
instance of: scholarly article
title: Lacunary directional maximal function on the Heisenberg group
zbMATH Open document ID: 1077.42014
author: Joonil Kim
published in: Mathematical Research Letters
publication date: 2005-04-05T00:00:00Z
review text: Let \(K=(k_1,k_2,k_3)\in{\mathbb Z}^3\) and \(v_K=(2^{k_1}, 2^{k_2}, 2^{k_3}).\) Let \(\mathbb H^1\) be the Heisenberg group identified with \({\mathbb R}^2\times {\mathbb R}^1\) endowed with the group multiplication: \[ (p,q,t)\cdot(p',q',t')=(p+p',q+q', t+t'+2(p'\cdot q-p\cdot q')). \] For any \(f\in L^1_{\text{ loc}}(\mathbb H^1)\), define \[ {\mathcal M}^{\text{ lac}}f(x)=\sup_{K\in {\mathbb Z}^3,\, r>0} \frac 1{2r}\int^r_{-r}f(x\cdot(tv_K)^{-1})\,dt. \] The author proves that for \(1<p\leq\infty\), \({\mathcal M}^{\text{lac}}\) is a bounded operator on \(L^p(\mathbb H^1)\) with an operator norm depending on \(p\). The proof is based on the group Fourier transform and the angular Littlewood-Paley decompositions in the Heisenberg group by the author in [Duke Math. J. 114, No. 3, 555--593 (2002; Zbl 1012.42013)].
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 2152363
zbMATH Keywords: Heisenberg group
zbMATH Keywords: lacunary directional maximal function
zbMATH Keywords: boundedness
reviewed by: Dachun Yang
MaRDI profile type: Publication
DOI: 10.4310/MRL.2004.V11.N5.A5
description: scientific article; zbMATH DE number 2152363