Vector valued q-variation for differential operators and semigroups I

Hong, Guixiang; Ma, Tao

doi:10.1007/s00209-016-1756-0

Cited by 42 publications

(36 citation statements)

References 41 publications

(72 reference statements)

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“…Proof of the Theorem 1.2. We are going to use the same arguments developed in the proof of Theorem 1.1, together with some results in [14,23].…”

Section: ]mentioning

confidence: 99%

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Variation and oscillation for harmonic operators in the inverse Gaussian setting

Almeida¹

2022

CPAA

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<p style='text-indent:20px;'>We prove variation and oscillation <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-boundedness properties for weighted difference involving the semigroups under consideration.</p>

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“…Proof of the Theorem 1.2. We are going to use the same arguments developed in the proof of Theorem 1.1, together with some results in [14,23].…”

Section: ]mentioning

confidence: 99%

“…Classical families of operators have been considered. For instance: semigroups of operators ([5, 16, 20, 29, 30]), differential operators ( [23,32]), singular integrals ( [14,15,22,24,32,33]), multipliers ( [17]), and Fourier series ( [18,36]).…”

mentioning

confidence: 99%

Variation and oscillation for harmonic operators in the inverse Gaussian setting

Almeida¹

2022

CPAA

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“…In scalar-valued harmonic analysis, we can realize this passage from the dyadic setting to the continuous one by dealing with issues such as rapidly decreasing tails or using the Vitali covering lemma. In the case of vector-valued harmonic analysis, this passage requires deep understanding on the connection between martingale theory and harmonic analysis as done in [4] [5] [6] [18] [19] [62] [38] etc. In noncommutative harmonic analysis, in addition to the idea or the techniques developed in vector-valued theory, new idea, techniques or tools developed in noncommutative analysis are usually needed to realize this passage such as in [42], [21].…”

Section: Corollary 13 Let T Be a Continuous Czo On R N Satisfyingmentioning

confidence: 99%

An operator-valued T1 theory for symmetric CZOs

Hong

Liu

Mei

2020

Journal of Functional Analysis

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We provide a natural BMO-criterion for the L 2 -boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the L 2 -boundedness of the commutators [R j , b] whenever b belongs to the Bourgain's vector-valued BMO space, where R j is the j-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.

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“…We refer to [12][13][14][15]20] for more information on the development of noncommutative harmonic analysis. We also refer to [3,11,[16][17][18]23]) for more information on related maximal inequalities.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative pointwise convergence of orthogonal expansions of several variables

Pang

Wang

2021

Mathematische Nachrichten

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In this paper, the boundedness of the maximal function for an operator-valued weighted 1 space on the unit sphere of ℝ +1 , in which the weight functions are invariant under finite reflection groups, is established. We use it to prove the boundedness of orthogonal expansions in ℎ-harmonics and this result applies to various methods of summability. Furthermore, we obtain the corresponding pointwise convergence theorems. At last, we give some results in the special reflection group ℤ 2 .

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Vector valued q-variation for differential operators and semigroups I

Cited by 42 publications

References 41 publications

Variation and oscillation for harmonic operators in the inverse Gaussian setting

Variation and oscillation for harmonic operators in the inverse Gaussian setting

An operator-valued T1 theory for symmetric CZOs

Noncommutative pointwise convergence of orthogonal expansions of several variables

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