Hausdorff operators on the power weighted Hardy spaces

Ruan, Jiaying; Fan, Dashan

doi:10.1016/j.jmaa.2015.07.062

Cited by 34 publications

(21 citation statements)

References 26 publications

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“…This class of operators contains some important examples, such as Hardy operator, the Cesàro operator and its q-calculus version, and there adjoints. As mentioned in [3] the Riemann-Liouville fractional integral and the Hardy-Littlewood-Polya operator can also be reduced to the Hausdorff operator, and as was noted in [5] in the one-dimensional case the Hausdorff operator is closely related to a Calderón-Zygmund convolution operator, too.…”

Section: Introductionmentioning

confidence: 75%

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Миротин

2019

Forum Mathematicum

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AbstractHausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}. Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

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Section: Introductionmentioning

confidence: 75%

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Миротин

2019

Forum Mathematicum

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“…Remark 6. The result by Ruan and Fan [26,Theorem 1.3] shows that for K = R the above condition of boundedness for H can not be sharpened in general.…”

Section: Finite-dimensional Spaces Over Locally Compact Division Ringsmentioning

confidence: 99%

“…This class of operators contains some important examples, such as Hardy operator, adjoint Hardy operator, the Cesàro operator. As mentioned in [26] the Riemann-Liouville fractional integral and the Hardy-Littlewood-Polya operator can also be reduced to the Hausdorff operator. As was noted in [4] the Hausdorff operator is closely related to a Calderón-Zygmund convolution operator, too.…”

Section: Introductionmentioning

confidence: 99%

Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups

Миротин

2019

Journal of Mathematical Analysis and Applications

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“…where A(y) is an n-th order square matrix, which satisfy det A(y) = 0 almost everywhere in the support of Φ ∈ L 1 loc (R n ). Using duality approach, Lerner and Liflayand [22] obtained the boundedness of H Φ,A on real Hardy space H 1 (R n ), after which the same problem was reconsidered in [7,23,31].…”

Section: Introductionmentioning

confidence: 99%