The Hodge–de Rham Laplacian and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math>-boundedness of Riesz transforms on non-compact manifolds

Chen, Peng; Magniez, Jocelyn; Ouhabaz, El Maati

doi:10.1016/j.na.2015.05.010

Cited by 13 publications

(16 citation statements)

References 26 publications

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“…It has to be compared with [18] where a negative answer for (G p ) (and so (R p )) is given for p > ν if the operator L has a positive ground state function.…”

Section: Examples and Applicationsmentioning

confidence: 99%

Riesz transforms through reverse Hölder and Poincaré inequalities

Bernicot

Frey

2016

Math. Z.

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Abstract. We study the boundedness of Riesz transforms in L p for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L p for p in some interval [2, 2 + ε), and that L p gradient estimates for the semigroup imply boundedness of the Riesz transform in L q for q ∈ [2, p). This improves results of [7] and [6], where the stronger assumption of a Poincaré inequality and the assumption e −tL (1) = 1 were made. The Poincaré inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.

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“…It has to be compared with [18] where a negative answer for (G p ) (and so (R p )) is given for p > ν if the operator L has a positive ground state function.…”

Section: Examples and Applicationsmentioning

confidence: 99%

Riesz transforms through reverse Hölder and Poincaré inequalities

Bernicot

Frey

2016

Math. Z.

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“…Now if the semigroup (e −t − → ∆ ) t≥0 is uniformly bounded on L p (Λ 1 T * M ) (and so by duality on L p (Λ 1 T * M )) then (34) holds for all r ∈ (min(p, p ), 2]. It is known that the validity of (34) implies that L r (Λ 1 T * M ) and H r − → ∆ (Λ 1 T * M ) coincide and have equivalent norms (see Proposition 2.5 in [9] and references there). This implies that d * − → ∆ −1/2 is bounded from L r (Λ 1 T * M ) to L r (M ) for all r ∈ (min(p, p ), 2].…”

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confidence: 94%

“…The commutation property − → ∆d = d∆ gives the desired result. We refer to [12] for additional details and to [3], [7], [8], [9], [19], [24] and the references therein for further results on the Riesz transform on L p for p > 2.…”

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confidence: 99%

$$L^p$$-Estimates for the Heat Semigroup on Differential Forms, and Related Problems

Magniez

Ouhabaz

2019

J Geom Anal

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We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let − → ∆ k be the Hodge-de Rham Laplacian on differential k-forms with k ≥ 1. By the Bochner decomposition formulais not uniformly bounded on L p for any p = 2. We also prove the gradient estimate ∇e −t∆ p−p ≤ Ct − 1 p , where ∆ is the Laplace-Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2.

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“…These results have been subsequently extended into various directions. As a sample of the extensive literature on this topic, we mention [15,[44][45][46]56] (for the Witten Laplacian); see also [3,4,10,19,37,42,47,49,52,54] (for the Laplace-Beltrami operator), [17,31,51] (for the Hodge-de Rham Laplacian), and [11] (for sub-elliptic operators).…”

Section: Introductionmentioning

confidence: 99%