The Riesz transformation on conical varieties
type: Item
instance of: scholarly article
title: The Riesz transformation on conical varieties
zbMATH Open document ID: 0937.43004
author: Hong-Quan Li
published in: Journal of Functional Analysis
publication date: 2000-02-14T00:00:00Z
review text: Let \(N\) be a connected Riemannian manifold of dimension \(n -1\geq 1\). This paper is concerned with the investigation of the Riesz transform \(\nabla(-\Delta)^{-1/2}\) on the cone \(C(N)= R^+\times N\). When \(N\) is compact, let \(\lambda_1\) be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on \(N\) and \[ p_0 =\text{sup}\bigl\{p>1\mid p.\bigl(\tfrac n2-\sqrt{(\tfrac{n-2}{2})^2+\lambda_1}\bigr)<n\bigr\}. \] The author shows that \(\nabla(-\Delta)^{-1/2}\) is of weak type (1,1) and type \((p,p)\) for all \(1 <p <p_0\) and that it is not of type \((p,p)\) for any \(p >p_0\). When \(N\) is a complete noncompact Riemannian manifold, the author studies the behavior of the Riesz transform on the cone \(C(N)\) in several cases. For example, if \(N =G/K\) is a Riemannian symmetric space of noncompact type then \(\nabla(-\Delta)^{-1/2}\) is bounded on \(L^p(C(N))\) for all \(1< p < n\), while if \(N= R^{n-1}\) with \(n \geq 3\), then \(\nabla(- \Delta)^{-1/2}\) is not bounded on \(L^p(C(N))\) for any \(p>n\).
reviewed by: Fuliu Zhu
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1374985
zbMATH Keywords: heat kernel
zbMATH Keywords: Riesz transform
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Riemannian symmetric space
MaRDI profile type: Publication
full work available at URL: jfan.1999.3464
OpenAlex ID: W2074530224
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DOI: 10.1006/JFAN.1999.3464
description: scientific article; zbMATH DE number 1374985