Weak type (1, 1) of some operators for the Laplacian with drift

Li, Hongquan; Sjögren, Peter; Wu, Yurong

doi:10.1007/s00209-015-1555-z

Cited by 10 publications

(10 citation statements)

References 31 publications

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“…For the other three operators, we have the following endpoint estimates. We remark that the weak type (1, 1) of H 0 was obtained in [20,Theorem 2].…”

Section: Introductionmentioning

confidence: 75%

“…. , 0), which is no restriction; see [20]. Then dμ = e 2x1 dx, and it will be convenient to write points in…”

Section: Notation and Simple Factsmentioning

confidence: 99%

“…Their setting and results are actually more general. In [20], the authors and Y.-R. Wu showed that the first-order Riesz transform ∇(−Δ v ) −1/2 is of weak type (1, 1) in R (n,v) . Here and in the sequel, L p and weak L p estimates in R (n,v) always refer to the measure μ.…”

Section: Introductionmentioning

confidence: 99%

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Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Li¹,

Sjögren

2020

Can. J. Math.-J. Can. Math.

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Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

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“…For the other three operators, we have the following endpoint estimates. We remark that the weak type (1, 1) of H 0 was obtained in [20,Theorem 2].…”

Section: Introductionmentioning

confidence: 75%

“…. , 0), which is no restriction; see [20]. Then dμ = e 2x1 dx, and it will be convenient to write points in…”

Section: Notation and Simple Factsmentioning

confidence: 99%

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Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Li¹,

Sjögren

2020

Can. J. Math.-J. Can. Math.

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“…See also [6] for other examples. Recently, the present authors studied Riesz transforms associated to the Laplacian with drift in [34,36] and [35]. Also notice that in the papers [36] and [35], which treat the Laplacian with drift in Euclidean space, the setting can be seen as the direct product of a Euclidean space and the simplest weighted manifold on the real line satisfying exponential volume growth with spectral gap.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the present authors studied Riesz transforms associated to the Laplacian with drift in [34,36] and [35]. Also notice that in the papers [36] and [35], which treat the Laplacian with drift in Euclidean space, the setting can be seen as the direct product of a Euclidean space and the simplest weighted manifold on the real line satisfying exponential volume growth with spectral gap. Let us finally observe that the setting of the present paper is natural in this context, since it consists of a sub-Laplacian with drift in a typical sub-Riemannian manifold, the Heisenberg group.…”

Section: Introductionmentioning

confidence: 99%

Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

Sjögren

2021

J Fourier Anal Appl

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In the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

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On non-centered maximal operators related to a non-doubling and non-radial exponential measure

et al. 2023

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We investigate mapping properties of non-centered Hardy–Littlewood maximal operators related to the exponential measure $$d\mu (x) = \exp (-|x_1|-\cdots -|x_d|)dx$$ d μ ( x ) = exp ( - | x 1 | - ⋯ - | x d | ) d x in $${\mathbb {R}}^d$$ R d . The mean values are taken over Euclidean balls or cubes ($$\ell ^{\infty }$$ ℓ ∞ balls) or diamonds ($$\ell ^1$$ ℓ 1 balls). Assuming that $$d \ge 2$$ d ≥ 2 , in the cases of cubes and diamonds we prove the $$L^p$$ L p -boundedness for $$p>1$$ p > 1 and disprove the weak type (1, 1) estimate. The same is proved in the case of Euclidean balls, under the restriction $$d \le 4$$ d ≤ 4 for the positive part.

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Weak type (1, 1) of some operators for the Laplacian with drift

Cited by 10 publications

References 31 publications

Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

On non-centered maximal operators related to a non-doubling and non-radial exponential measure

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