The ρ-variation of the Hermitian Riesz transform

Crescimbeni, Raquel; Martín-Reyes, F. J.; Torre, A. de la; Torrea, José L.

doi:10.1007/s10114-010-9122-3

Cited by 24 publications

(13 citation statements)

References 11 publications

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“…In [23, p. 516] it was proved that if V is a nonnegative polynomial, then V ∈ B q , for every 1 < q < ∞. Then, as special cases of our results appear the corresponding ones to the Hermite operator H = −∆ + |x| 2 ([9] and [10]). This paper is organized as follows.…”

Section: Introduction and Main Resultsmentioning

confidence: 70%

L p -boundedness properties of variation operators in the Schrödinger setting

2012

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Section: Introduction and Main Resultsmentioning

confidence: 70%

L p -boundedness properties of variation operators in the Schrödinger setting

2012

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“…1 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China. 2 School of Mathematics and Statistics, Yili Normal College, Yining, Xinjiang 835000, China.…”

Section: Competing Interestsmentioning

confidence: 99%

Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions

Zhang

2015

J Inequal Appl

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This paper is devoted to investigating the boundedness of the oscillation and variation operators for the commutators generated by Calderón-Zygmund singular integrals with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces in dimension 1. Certain criterions of boundedness are given. As applications, the weighted (L p , L q )-estimates for the oscillation and variation operators on the iterated commutators of Hilbert transform and Hermitian Riesz transform, the (L p ,∧ (β-1/p) )-bounds as well as the endpoint estimates for the oscillation and variation operators of the corresponding first order commutators are established. MSC: Primary 42B20; 42B25; secondary 47G10

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“…In 2004, Gillespie and Torrea [9] established the L p (R, w(x)dx) bounds for O(H) and V (H) with > 2, 1 < p < ∞ and w ∈ A p (the Muckenhoupt weights class) (also see [10,14] for the related investigations). Later on, Crescimbeni et al [5] proved that O(H) and V (H) with ρ > 2 map L 1 (R, w(x)dx) into L 1,∞ (R, w(x)dx) for w ∈ A 1 . In particular, Ma et al [21,22] presented the weighted oscillation and variation inequalities for differential operators and Calderón-Zygmund singular integrals.…”

Section: Introductionmentioning

confidence: 99%

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

Liu

Jhang

et al. 2019

Mathematics

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We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.

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The ρ-variation of the Hermitian Riesz transform

Cited by 24 publications

References 11 publications

L p -boundedness properties of variation operators in the Schrödinger setting

L p -boundedness properties of variation operators in the Schrödinger setting

Oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

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