##
**Riesz transform and perturbation.**
*(English)*
Zbl 1122.58014

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 result in \(\mathbb{R}^d\), only differing in the additional assumption that the Riesz transform of \(H\) is bounded in \(L^{p'}\) for \(p'\in (p_0',2)\), where \(p_0'\) is the conjugate exponent of \(p_0\).

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 result in \(\mathbb{R}^d\), only differing in the additional assumption that the Riesz transform of \(H\) is bounded in \(L^{p'}\) for \(p'\in (p_0',2)\), where \(p_0'\) is the conjugate exponent of \(p_0\).

Reviewer: Nils Ackermann (México, D.F.)

### MSC:

58J37 | Perturbations of PDEs on manifolds; asymptotics |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

47F05 | General theory of partial differential operators |

47B44 | Linear accretive operators, dissipative operators, etc. |

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\textit{T. Coulhon} and \textit{N. Dungey}, J. Geom. Anal. 17, No. 2, 213--226 (2007; Zbl 1122.58014)

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