scholarly journals On the Structure of Normal Hausdorff Operators on Lebesgue Spaces

2019 ◽  
Vol 53 (4) ◽  
pp. 261-269 ◽  
Author(s):  
A. R. Mirotin
Keyword(s):  
Lebesgue Spaces ◽  
Hausdorff Operators

scholarly journals On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

2020 ◽  
Vol 32 (1) ◽  
pp. 111-119 ◽  
Author(s):  
Adolf R. Mirotin
Keyword(s):  
Present Article ◽  
Spectral Representation ◽  
Research Field ◽  
Hausdorff Operator ◽  
Lebesgue Spaces ◽  
Summation Methods ◽  
Hausdorff Operators ◽  
Active Research ◽  
Matrix Symbol

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.

scholarly journals Boundedness of Multidimensional Dunkl-Hausdorff Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Radouan Daher ◽  
Faouaz Saadi
Keyword(s):  
Hardy Space ◽  
Sufficient Conditions ◽  
Hausdorff Operator ◽  
Lebesgue Spaces ◽  
Dunkl Operators ◽  
Weighted Lebesgue Spaces ◽  
Hausdorff Operators

In the present paper, we introduce the multidimensional Dunkl-Hausdorff operator ℋκ and we give simple sufficient conditions so that these operators be bounded on the weighted lebesgue spaces Lκpℝn and in the Hardy space Hκ1ℝn associated with the Dunkl operators. We also determine the Dunkl-Hausdorff operator ℋκ∗ that is adjoint to ℋκ.

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

2018 ◽  
Vol 61 (9) ◽  
pp. 1647-1664 ◽  
Author(s):  
Jiecheng Chen ◽  
Jiawei Dai ◽  
Dashan Fan ◽  
Xiangrong Zhu
Keyword(s):  
Hardy Spaces ◽  
Lebesgue Spaces ◽  
Hausdorff Operators

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators

scholarly journals ON THE COMPLEXITY OF CLASSIFYING LEBESGUE SPACES

2020 ◽  
pp. 1-35
Author(s):  
TYLER A. BROWN ◽  
TIMOTHY H. MCNICHOLL ◽  
ALEXANDER G. MELNIKOV
Keyword(s):  
Lebesgue Spaces

scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

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