An \((n+1)\)-fold Marcinkiewicz multiplier theorem on the Heisenberg group
type: Item
instance of: scholarly article
zbMATH Open document ID: 0979.43004
DOI: 10.1017/S0004972700019092
author: A. J. Fraser
published in: Bulletin of the Australian Mathematical Society
publication date: 2002-02-19T00:00:00Z
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 42A45
zbMATH DE Number: 1599685
zbMATH Keywords: Marcinkiewicz multiplier
zbMATH Keywords: Heisenberg group
zbMATH Keywords: convolution operator
zbMATH Keywords: convolution kernels
MaRDI profile type: Publication
cites work: Q5607484
cites work: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II
cites work: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I
cites work: Fourier analysis on the Heisenberg group
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cites work: Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains
cites work: Calderón-Zygmund operators on product spaces
title: An \((n+1)\)-fold Marcinkiewicz multiplier theorem on the Heisenberg group
review text: In this paper the Marcinkiewicz multiplier theorem is proved on the Heisenberg group. Let \(\mathbb{H}_n= \mathbb{C}^n \times\mathbb{R}\) denote the \((2n+1)\)-dimensional Heisenberg group with the multiplication law NEWLINE\[NEWLINE(z,t) (w,s)= (z+w,t+s+2 {\mathfrak I}z \cdot\overline \omega).NEWLINE\]NEWLINE Using the coordinate \(h= (z,t)= (x+iy,t)\) for points in \(\mathbb{H}_n\), the left-invariant vector fields NEWLINE\[NEWLINEX_j= {\partial\over \partial x_j} +2y_j{\partial \over\partial t}, \quad Y_j= {\partial\over \partial y_j}- 2x_j{\partial \over\partial t},\quad j=1,2,\dots, n, \text{ and }T= {\partial\over \partial t}NEWLINE\]NEWLINE form a basis for the Lie algebra of \(\mathbb{H}_n\). The partial sub-Laplacians \({\mathcal L_1}, \dots, {\mathcal L}_n\) are given by \({\mathcal L}_j= -{1\over 4}(X_j^2+Y_j^2)\), \(j=1,\dots,n\). The operators \({\mathcal L}_1, \dots, {\mathcal L}_n\) and \(iT\) form a family of commuting self-adjoint operators and, for \(m\in L^\infty ((\mathbb{R}^+)^n \times\mathbb{R})\), we can define the joint spectral multiplier operator \(m({\mathcal L}_1, \dots, {\mathcal L}_n,iT)\), which is also given as a convolution operator with a kernel in the distribution space \({\mathcal S}'(\mathbb{H}_n)\). For \(1<p<\infty\) the author establishes the boundedness on \(L^p(\mathbb{H}_n)\) of the multiplier operators \(m({\mathcal L}_1,\dots, {\mathcal L}_n,iT)\) if \(m\) satisfies an \((n+1)\)-fold Marcinkiewicz-type condition NEWLINE\[NEWLINE\bigl|(\xi_1\partial_{\xi_1})^{i_1} \cdots(\xi_n \partial_{\xi_n})^{i_n} (\eta \partial_\eta)^j m(\xi,\eta) \bigr|\leq C_{ij}NEWLINE\]NEWLINE for all \(i=(i_1, \dots, i_n)\), \(j\leq 9n\). Furthermore, the author establishes regularity and cancellation conditions which the convolution kernels of these Marcinkiewicz-type multipliers \(m({\mathcal L}_1, \dots, {\mathcal L}_n,iT)\) satisfy.
reviewed by: Koichi Saka
description: scientific article; zbMATH DE number 1599685