Analysis of the Laplacian on the complete Riemannian manifold

Strichartz, Robert S.

doi:10.1016/0022-1236(83)90090-3

Cited by 470 publications

(399 citation statements)

References 22 publications

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“…It is well known that for each nonnegative integer k, the restriction of L to L 2 (Λ k C M ) is a self adjoint operator [28]. In particular, the restriction of L to L 2 (Λ 0 C M ) coincides with L. Furthermore, it is known that the restriction of L to 1-forms is nonnegative.…”

Section: Notation Background Materials and Preliminary Resultsmentioning

confidence: 99%

Estimates for functions of the Laplacian on manifolds with bounded geometry

Mauceri

Meda

Vallarino

2009

Mathematical Research Letters

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Abstract. In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M and by D the operator √ L − b, where b denotes the bottom of the spectrum of L. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded, even holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hörmander type conditions of appropriate order at infinity, then the operator m(D) extends to an operator of weak type 1.This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M , but without any assumption on the decay of the heat kernel.

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Section: Notation Background Materials and Preliminary Resultsmentioning

confidence: 99%

Estimates for functions of the Laplacian on manifolds with bounded geometry

Mauceri

Meda

Vallarino

2009

Mathematical Research Letters

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“…One useful ingredient in this regard is (53) Ω |u| 2 dx ≤ C Ω {|∇u| 2 + W |u| 2 } dx, itself a version of Poincaré's inequality. When used in conjunction with (46) and (49), this readily yields…”

Section: Layer Potentials Depending On a Parametermentioning

confidence: 99%

“…Note that in our paper we work exclusively with manifolds of bounded geometry. The papers [42] and [49] are a good introduction to some basic results on the analysis on non-compact manifolds. Throughout the paper, a classical pseudodifferential operator P will be called elliptic if its principal symbols is invertible outside the zero section.…”

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confidence: 99%

Boundary value problems and layer potentials on manifolds with cylindrical ends

Mitrea

Nistor

2007

Czech Math J

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Abstract. We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the noncompactness of the boundary, which prevents us from using the standard characterization of Fredholm and compact (pseudo-)differential operators between Sobolev spaces. Our approach, which involves the study of layer potentials depending on a parameter on compact manifolds as an intermediate step, yields the invertibility of the relevant boundary integral operators in the global, noncompact setting, which is rather unexpected. As an application, we prove the well-posedness of the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichletto-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are "almost translation invariant at infinity," a calculus that is closely related to Melrose's b-calculus [29,28], which we study in this paper. The proof of the convergence of the layer potentials and of the existence of the Dirichlet-to-Neumann map are based on a good understanding of resolvents of elliptic operators that are translation invariant at infinity.

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“…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.…”

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confidence: 99%

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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Analysis of the Laplacian on the complete Riemannian manifold

Cited by 470 publications

References 22 publications

Estimates for functions of the Laplacian on manifolds with bounded geometry

Estimates for functions of the Laplacian on manifolds with bounded geometry

Boundary value problems and layer potentials on manifolds with cylindrical ends

Riesz transform, Gaussian bounds and the method of wave equation

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