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Thalmaier, Anton; Wang, Feng‐Yu

doi:10.1023/a:1026310604320

Cited by 7 publications

(7 citation statements)

References 20 publications

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“…Thus to investigate whether these L p -Calderón-Zygmund inequalities hold are reduced to the study of conditions for boundedness of the classical Riesz transform d(∆ µ + σ) −1/2 on functions and boundedness of the covariant Riesz transform ∇(∆ (1) µ + σ) −1/2 on one-forms in L p -sense. Therefore, in [5] combining this argument with the result in [29] yields that the L p -Calderón-Zygmund inequalities hold for 1…”

Section: Introductionmentioning

confidence: 71%

“…The Riesz transform ∇∆ −1/2 f , introduced by Strichartz [27] on Euclidean space, has been investigated in many subsequent papers, see e.g., [8,2,3,4] and the references therein, and has been further extended to Riemannian manifolds, e.g. [1,11,12,13,23,29]. Since the Riesz transform is bounded in L 2 (M), by the interpolation theorem, the weak (1, 1) property already implies L p (M)-boundedness for p ∈ (1,2].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we aim to study covariant Riesz transforms ∇(∆ (k) + σ) −1/2 on Riemannian vector bundles for p ∈ (1,2] which poses comparably more difficulties to deal with. This question has already been addressed by the second and third named author in [29], where the authors adopted the method of Coulhon and Duong [11] relying on the doubling volume property, Li-Yau type heat kernel upper bounds and derivative estimates of the heat kernel. Note that the approach in [29] is of stochastic nature and derivative estimates for the heat kernel are deduced from derivative formulae for semigroups on vector bundles, by means of the methodology of Driver and the second named author [15].…”

Section: Introductionmentioning

confidence: 99%

“…This question has already been addressed by the second and third named author in [29], where the authors adopted the method of Coulhon and Duong [11] relying on the doubling volume property, Li-Yau type heat kernel upper bounds and derivative estimates of the heat kernel. Note that the approach in [29] is of stochastic nature and derivative estimates for the heat kernel are deduced from derivative formulae for semigroups on vector bundles, by means of the methodology of Driver and the second named author [15]. In [15] estimates of certain functionals of Brownian motion with respect to the Wiener measure are required; nevertheless pointwise estimate for the heat kernel e −t∆ (k) (x, y) and the derivative estimate of heat kernel ∇e −t∆ (k) (x, y) can be obtained from such general derivative formulas, where ∆ (k) is the unique self-adjoint realization of the Hodge-de Rham Laplacian acting on k-forms under explicit curvature condition, see [5].…”

Section: Introductionmentioning

confidence: 99%

“…As explained in [29], it is difficult to follow the corresponding argument in [11] directly concerning derivatives of heat kernel on vector bundle, since the heat kernel e −t∆ (k) (x, y) is linear operator on a vector bundle E → M from E y to E x . In this paper, we aim to overcome this difficulty by using a different approach, that is the Weitzenböck formula and the Gaussian type estimates for some Schrödinger heat kernels on manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Covariant Riesz transform on differential forms for $$1<p\le 2$$

Cheng,

Thalmaier,

Wang

2023

Calc. Var.

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In this paper, we study L p -boundedness (1 < p ≤ 2) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of L p -boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for 1 < p ≤ 2 on weighted manifolds is derived under the curvaturedimension condition as hypothesis.

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Covariant Riesz transform on differential forms for $$1<p\le 2$$

Cheng,

Thalmaier,

Wang

2023

Calc. Var.

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Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

2022

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With $$\vec {\Delta }_j\ge 0$$ Δ → j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and $$\nabla $$ ∇ the Levi-Civita covariant derivative, we prove among other things a Gaussian heat kernel bound for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted $$L^p$$ L p -bound for the heat kernel of $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, that $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j is bounded in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ , if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of $$\nabla $$ ∇ and $$\vec {\Delta }_j$$ Δ → j ) for $$\mathrm {e}^{ -t\vec {\Delta }_j }$$ e - t Δ → j for small times, if the j-th degree Bochner-Lichnerowicz potential $$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$ V j = Δ → j - ∇ † ∇ of M is bounded from below (where $$V_1=\mathrm {Ric}$$ V 1 = Ric ), which is shown to fail for large time, if $$V_j$$ V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$ ∇ ( Δ → j + κ ) - 1 / 2 in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ (which we prove for $$1\le p\le 2$$ 1 ≤ p ≤ 2 ), and explain its implications to geometric analysis, such as the $$L^p$$ L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j .

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Scattering Theory for the Hodge Laplacian

Baumgarth

2022

J Geom Anal

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We prove using an integral criterion the existence and completeness of the wave operators 𝑊 ± (Δ𝑔,ℎ ) corresponding to the Hodge-Laplacians Δ (𝑝) 𝜈 acting on differential 𝑝-forms, for 𝜈 ∈ {𝑔, ℎ}, induced by two quasi-isometric Riemannian metrics 𝑔 and ℎ on an complete open smooth manifold 𝑀. In particular, this result provides a criterion for the absolutely continuous spectra 𝜎 ac (Δ𝜈 to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion only requires local curvature bounds and some upper local control on the heat kernel acting on functions, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

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Cited by 7 publications

References 20 publications

Covariant Riesz transform on differential forms for $$1<p\le 2$$

Covariant Riesz transform on differential forms for $$1<p\le 2$$

Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

Scattering Theory for the Hodge Laplacian

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