Riesz transform on manifolds and heat kernel regularity

Auscher, Pascal; Coulhon, Thierry; Duong, Xuan Thinh; Hofmann, Steve

doi:10.1016/j.ansens.2004.10.003

Cited by 219 publications

(437 citation statements)

References 90 publications

(164 reference statements)

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“…Finally in Section 5 we give the proof of Theorems 3.1 and 3.2. The author thanks S. Hofmann for pointing out the relevance of the results in [1], [2]. The author also would like to thank the referee for several valuable comments.…”

Section: In Particular ∇(L) −1/2 Is Bounded On L P (ωDx) If and Onlmentioning

confidence: 99%

Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Shen¹

2005

Annales De L’institut Fourier

144

163

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Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied.

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Section: In Particular ∇(L) −1/2 Is Bounded On L P (ωDx) If and Onlmentioning

confidence: 99%

Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Shen¹

2005

Annales De L’institut Fourier

144

163

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“…Proof: The proof of the estimate for the first term is analogous to [23] (this kind of estimate originated in Gaffney's work [27]) and [24], Lemma 7, whereas the second term can be estimated by the same method as in [2], estimate (3.1) p. 930.…”

Section: Remark 34 Note That With the Same Notations Ifmentioning

confidence: 99%

“…Fortunately, there is a weaker notion of Gaussian decay, which holds on any complete Riemannian manifold, namely the notion of L 2 off-diagonal estimates, as introduced by Gaffney [27]. This notion has already proved to be a good substitute of Gaussian estimates for such questions as the Kato square root problem or L p -bounds for Riesz transforms when dealing with elliptic operators (even in the Euclidean setting) for which Gaussian estimates do not hold (see [1,4,9] in the Euclidean setting, and [2] in a complete Riemannian manifold). We show in the present work that a theory of Hardy spaces of differential forms can be developed under such a notion.…”

Section: 2)mentioning

confidence: 99%

Hardy Spaces of Differential Forms on Riemannian Manifolds

2007

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Abstract. Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.

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“…We refer the reader to the paper [3] for general results and its section 1.3 for the state of the art about Riesz transform L p boundedness. Here we mention that for the Laplacian, T ∈ L(L p ) if p ∈ (1, 2] in a quite general setting (see Coulhon-Duong [6]); in other words low ranges of p seem less sensitive to the geometry.…”

Section: Introductionmentioning

confidence: 99%

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

2008

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Abstract. Let M • be a complete noncompact manifold and g an asymptotically conic manifold on M • , in the sense that M • compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M . A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1 ; such manifolds have an end that can be identified with R n \ B(R, 0) in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2 ) −1 where P = ∆g + V is the sum of the positive Laplacian associated to g and a real potential function V that is smooth on M and vanishes to third order at the boundary (i.e. decays to third order at infinity on M • ). We show that on a blown up version of M 2 × [0, k 0 ] the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on L p (M • ) for 1 < p < n, and that this range is optimal if V ≡ 0 or if M has more than one end. The result with V ≡ 0 is new even when M • = R n , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 < p < pmax where pmax > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary ∂M .Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In a follow-up paper we shall analyze the same situation in the presence of zero modes and zero-resonances.

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Riesz transform on manifolds and heat kernel regularity

Cited by 219 publications

References 90 publications

Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators

Hardy Spaces of Differential Forms on Riemannian Manifolds

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

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