Endpoint Estimates for Commutators of Riesz Transforms Associated With Schrödinger Operators

Li, Pengtao; Peng, Lizhong

doi:10.1017/s0004972710000390

Cited by 16 publications

(11 citation statements)

References 7 publications

(10 reference statements)

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“…In this paper we are interested in the weighted Hardy space estimates for R , which are also the weighted endpoint estimates. It is noted that our main results generalize Theorem 2.7 and Theorem 4.1 in [3] to the weighted case and the function that we consider belongs to a larger class than the classical BMO space.…”

Section: Introductionsupporting

confidence: 65%

“…Recently, some scholars have investigated the boundedness of the commutators generated by a BMO function and Riesz transforms associated with the Schrödinger operator (cf. [2][3][4][5][6][7][8]). It follows from [9] that Riesz transform associated with the Schrödinger operator is not a Calderón-Zygmund operator if the potential ∈ (( /2) < < ).…”

Section: Introductionmentioning

confidence: 99%

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Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Liu

Sheng

Wang

2013

Abstract and Applied Analysis

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Let = −Δ + be a Schrödinger operator, where Δ is the laplacian on R and the nonnegative potential belongs to the reverse Hölder class 1 for some 1 ≥ ( /2). Assume that ∈ 1 (R ). Denote by 1 ( ) the weighted Hardy space related to the Schrödinger operator = −Δ + . Let R = [ , R] be the commutator generated by a function ∈ BMO (R ) and the Riesz transform R = ∇(−Δ + ) −(1/2) . Firstly, we show that the operator R is bounded from 1 ( ) into 1 weak ( ). Secondly, we obtain the endpoint estimates for the commutator [ , R]. Namely, it is bounded from the weighted Hardy space 1 ( ) into 1 weak ( ).

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Section: Introductionsupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Liu

Sheng

Wang

2013

Abstract and Applied Analysis

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“…As the Riesz transforms R j = ∂ x j L −1/2 are of weak type (1, 1) (see [30]), the following can be seen as a consequence of Proposition 3.1 (see also [32]).…”

Section: Statement Of the Resultsmentioning

confidence: 97%

Endpoint estimates for commutators of singular integrals related to Schrödinger operators

2015

Rev. Mat. Iberoam.

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Let L = −∆ + V be a Schrödinger operator on R d , d ≥ 3, where V is a nonnegative potential, V = 0, and belongs to the reverse Hölder class RH d/2 . In this paper, we study the commutators [b, T ] for T in a class K L of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T ∈ K L , we prove that there exists a bounded subbilinearThe subbilinear decomposition (1) allows us to explain why commutators with the fundamental operators are of weak type (H 1 L , L 1 ), and when a commutator [b, T ] is of strong type (H 1 L , L 1 ). Also, we discuss the H 1 L -estimates for commutators of the Riesz transforms associated with the Schrödinger operator L. Contents2010 Mathematics Subject Classification. Primary: 42B35, 35J10 Secondary: 42B20.

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“…In recent years, many scholars have investigated the singular integral operators related to Schrödinger operators L 1 and their commutators, see, for example, [17], [9], [7], [12], [1], [10], [16], [13], [15] and their references. Especially, when b ∈ BMO, Guo, Li and Peng [7] investigated the boundedness of the commutators of the Riesz transform ∇L −1/2 1 .…”

Section: Introductionmentioning

confidence: 99%

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

et al. 2016

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Let L 1 = −∆ + V be a Schrödinger operator and let L 2 = (−∆) 2 + V 2 be a Schrödinger type operator on R n (n 5), where V = 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s n/2. The Hardy type space H 1 L2 is defined in terms of the maximal function with respect to the semigroup {e −tL2 } and it is identical to the Hardy space H 1 L1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator R b = bRf − R(bf ) generated by the Riesz transform R = ∇ 2 L −1/2 2 , where b ∈ BMO θ (̺), which is larger than the space BMO(R n ). Moreover, we prove that R b is bounded from the Hardy space H 1 L2 (R n ) into weak L 1 weak (R n ).

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Endpoint Estimates for Commutators of Riesz Transforms Associated With Schrödinger Operators

Cited by 16 publications

References 7 publications

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

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