Riesz transform and related inequalities on non‐compact Riemannian manifolds

Coulhon, Thierry; Duong, Xuan Thinh

doi:10.1002/cpa.3040

Cited by 85 publications

(118 citation statements)

References 38 publications

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“…Theorem 10 generalizes results described in [10,11] to a very natural and somehow optimal setting. Proof.…”

Section: Hodge Laplacian and Riesz Transform For P >supporting

confidence: 70%

“…The proofs from [8,14,22] do not easily generalize to the De Rham-Hodge Laplacian setting. It turns out that the understanding of the behavior of the heat kernel generated by the Hodge Laplacian is a useful tool in the study of the L p boundedness of the Riesz transform for p > 2 (see [10] and [11]). In Theorem 10 we describe a natural generalization of the main result from [10] and [11,Theorem 5.5].…”

Section: Introductionmentioning

confidence: 99%

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Riesz transform, Gaussian bounds and the method of wave equation

2004

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Abstract. For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p > 2 . IntroductionLet ∆ = − n i=1 ∂ 2 i be the standard Laplace operator acting on R n . Then the corresponding Riesz transform is defined by the formula ∂ j ∆ −1/2 . The L p continuity of the Riesz transform for all p ∈ (1, ∞) is one of the most important and celebrated results in analysis. Papers devoted to the study of the Riesz transform and its generalizations are too numerous to list here. Hence we would like to mention only a few most relevant works [1,3,9,10,17,18,21,26,27,33,34,35,38,39,42].The operator ∇L −1/2 , where ∇ is the gradient and L is the Laplace-Beltrami operator on a Riemannian manifold M, is a natural generalization of the classical Riesz transform. L 2 boundedness of the Riesz transform ∇L −1/2 is a consequence of the equality ∇f L 2 = L 1/2 f L 2 , which is actually the definition of the operator L. In [42] Strichartz asked whether one could extend L p continuity of the classical Riesz transform to the setting of Laplace-Beltrami operators described above. An answer to this question was given in [9] for 1 ≤ p ≤ 2. In [9] Coulhon and Duong proved that if M is a complete Riemannian manifold which satisfies the doubling volume property (see Assumption 1), L is the Laplace-Beltrami operator on M and the heat kernel corresponding to L satisfies the Gaussian bounds then the Riesz transform ∇L −1/2 is of weak type (1, 1) and so bounded on L p for all p ∈ (1, 2]. Note that (∂ j ∆ −1/2 ) * = −∂ j ∆ −1/2 so the boundedness of the standard Riesz transform ∂ j ∆ −1/2 for p ∈ (1, 2] implies continuity of the standard Riesz transform for all p ∈ (1, ∞). Surprisingly in general the Riesz transform corresponding to the Laplace-Beltrami operator 1991 Mathematics Subject Classification. 42B20.

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“…Theorem 10 generalizes results described in [10,11] to a very natural and somehow optimal setting. Proof.…”

Section: Hodge Laplacian and Riesz Transform For P >supporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Riesz transform, Gaussian bounds and the method of wave equation

2004

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“…Such an idea, which originates from [51], has been formalised in [35] for boundedness results in the range 1 < p < 2 and is actually used in [18] to derive Theorem 1.1. However, this method does not apply to our situation as p > 2; a duality argument would not help us either as we would have to make assumptions on the semigroup acting on 1-forms as explained in Section 1.4; this would bring us back to the state of the art in [20] (see [88] for this approach to the results in [20]). But recently, it was shown in [72] that this regularity property can be used for L p results in the range p > 2 by employing good-λ inequalities as in [41] for an ad hoc sharp maximal function; this may be seen as the basis to an L ∞ to BM O version of the L 1 to weak L 1 theory in [35].…”

Section: About Our Methodsmentioning

confidence: 99%

“…Third, we use very little of the differential structure on manifolds, and in particular we do not use the heat kernel on 1-forms as in [7], [8], or [20]. As a matter of fact, our method is quite general, and enables one to prove the L p boundedness of a Riesz transform of the form ∇L −1/2 as soon as the following ingredients are available:…”

Section: About Our Methodsmentioning

confidence: 99%

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Riesz transform on manifolds and heat kernel regularity

Auscher

Coulhon

Duong

et al. 2004

Annales Scientifiques de l’École Normale Supérieure

221

413

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On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p , pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d'exposants p.One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.MSC numbers 2000: 58J35, 42B20

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New function spaces of BMO type, the John‐Nirenberg inequality, interpolation, and applications

Duong

Yan

2005

Comm Pure Appl Math

Self Cite

218

319

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In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John-Nirenberg inequality holds on these spaces and they interpolate with L p spaces by the complex interpolation method. We then give applications on L pboundedness of singular integrals whose kernels do not satisfy the Hörmander condition.

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Riesz transform and related inequalities on non‐compact Riemannian manifolds

Cited by 85 publications

References 38 publications

Riesz transform, Gaussian bounds and the method of wave equation

Riesz transform, Gaussian bounds and the method of wave equation

Riesz transform on manifolds and heat kernel regularity

New function spaces of BMO type, the John‐Nirenberg inequality, interpolation, and applications

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