Riesz transform and perturbation
type: Item
instance of: scholarly article
title: Riesz transform and perturbation
zbMATH Open document ID: 1122.58014
DOI: 10.1007/BF02930721
author: Thierry Coulhon
author: Nick Dungey
published in: The Journal of Geometric Analysis
publication date: 2007-06-04T00:00:00Z
review text: Suppose that \(A\colon\mathbb{R}^d\to\mathbb{R}^{d\times d}\) is measurable and satisfies the ellipticity and boundedness estimate \[ C^{-1}| v| ^2\leq v^TA(x)v\leq C| v| ^2 \] with some \(C>0\). Let \(H\) denote the form closure of the divergence form operator \(-\text{div}A\nabla\) on \(L^p(\mathbb{R}^d)\). The Riesz transform of \(H\) is the operator \(\nabla H^{-1/2}\) on \(L^p\). It is known that the Riesz transform of \(H\) is bounded for \(p\in(1,2+\varepsilon)\), where \(\varepsilon>0\) depends on \(d\) and \(C\). The present article gives a perturbation type criterion to extend this result to higher values of \(p\) and to closed noncompact Riemannian manifolds. The first result pertains to the Euclidean case. Suppose one is given two divergence form operators \(H_0\) and \(H\), induced by the corresponding matrix functions \(A_0\) and \(A\) as above. Consider \(H\) as a perturbation of \(H_0\), assuming that \(A_0-A\in L^q\) for some \(q\in[1,\infty)\). If the Riesz transform of \(H_0\) is bounded in \(L^{p_0}\) for some \(p_0>2\), and if \(\nabla(I+H)^{-1/2}\) (the local Riesz transform of \(H\)) is bounded in \(L^p\) for all \(p\in(2,p_0)\), then the Riesz transform of \(H\) is bounded in \(L^p\) for all \(p\in(2,p_0)\). A generalization of this result to weighted \(L^p\) spaces, where the weights are positive, bounded, and bounded away from \(0\), leads to a theorem for a noncompact, connected smooth manifold \(M\) of dimension \(d\): Suppose that \(G_0\) and \(G\) are two Riemannian metrics on \(M\) with uniformly equivalent associated norms on the tangent spaces. Assume that \(G\) is a \(L^q\)-perturbation of \(G_0\) in a specified sense, for some \(q\in[1,\infty)\). Denote by \(H_0\) and \(H\) the positive Laplace operators associated with \(G_0\) and \(G\). Moreover, assume the norm estimate \[ \| e^{-tH_0}\| _{{\mathcal L}(L^1,L^\infty)}\leq C_1t^{-C_2},\qquad t\geq 1, \] with positive constants \(C_1,C_2\). Then there holds a theorem similar to the…
reviewed by: Nils Ackermann
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47B44
zbMATH DE Number: 5160936
zbMATH Keywords: Riesz Transform
zbMATH Keywords: Noncompact Closed Riemannian Manifold
zbMATH Keywords: Perturbation
MaRDI profile type: Publication
cites work: Q5489536
cites work: Riesz transform on manifolds and heat kernel regularity
cites work: Q4217284
cites work: Stability of weighted Laplace-Beltrami operators under \(L^ p\)-perturbation of the Riemannian metric
cites work: Riesz transforms for $1\le p\le 2$
cites work: Riesz transforms for \(p>2\)
cites work: Riesz transform and related inequalities on non‐compact Riemannian manifolds
cites work: Large time behavior of heat kernels on forms
cites work: Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds
cites work: Heat kernel on a manifold with a local Harnack inequality
cites work: \(L^p\) bounds for Riesz transforms and square roots associated to second order elliptic operators
cites work: Gradient estimate of the heat kernel on modified graphs
cites work: Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds
cites work: Bounds of Riesz transforms on \(L^p\) spaces for second order elliptic operators.
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description: scientific article; zbMATH DE number 5160936