About Riesz transforms on the Heisenberg groups
type: Item
instance of: scholarly article
title: About Riesz transforms on the Heisenberg groups
zbMATH Open document ID: 0859.22006
DOI: 10.1007/BF01444227
author: Thierry Coulhon
author: J. Zienkiewicz
author: Detlef Müller
published in: Mathematische Annalen
publication date: 1996-10-07T00:00:00Z
full work available at URL: https://eudml.org/doc/182993
review text: Let \(X_1,\dots,X_{2n}\), \(Z\) be the left invariant vector fields on the Heisenberg group \(H^n\) given by \(X_{2j-1}=\partial_{x_j}+2y_j\partial_z\), \(X_{2j}=\partial_{y_j}-2x_j\partial_z\) and \(Z=\partial_z\). Set \(\Delta=-\sum^{2n}_{i=1}X^2_i\) and consider the Riesz transforms \(R_i=X_i\Delta^{-1/2}\) for \(i\in\{1,\dots,2n\}\). Using an explicit formula for the heat kernel associated with the sublaplacian \(\Delta\) the authors prove that for all \(p\in]1,\infty[\) there exists a \(C_p>1\) such that \[ C^{-1}_p|f|_{L^p(H^n)}\leq \left|\left(\sum^{2n}_{i=1}|R_if|^2\right)^{1/2} \right|_{L^p(H^n)} \leq C_p|f|_{L^p(H^n)} \] uniformly for all \(n\in \mathbb{N}^*\) and \(f\in C^\infty_0(H^n)\). Note that \(C_p\) is independent of \(n\).
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 931963
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Riesz transforms
zbMATH Keywords: heat kernel
zbMATH Keywords: sublaplacian
MaRDI profile type: Publication
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OpenAlex ID: W2016673693
description: scientific article; zbMATH DE number 931963