Sharp estimates for Hardy operators on Heisenberg group

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite...

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Bibliographic Details
Published inFrontiers of Mathematics Vol. 11; no. 1; pp. 155 - 172
Main Authors WU, Qingyan, FU, Zunwei
Format Journal Article
LanguageEnglish
Published Beijing Higher Education Press 01.02.2016
Springer Nature B.V
Subjects
Boundary conditions
Estimates
Functions (mathematics)
Hardy operator
Hardy算子
Harmonic analysis
Heisenberg group
Inequalities
Inequality
Lie groups
M p weight
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Norms
Operators
Operators (mathematics)
p-范数
Research Article
Studies
Translations
不等式
估计
加权和
最佳常数
有界性
海森堡群
Hardy operator
22E25
Heisenberg group
weight
43A15
26D10
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Summary:In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on R, balls in Rn, or ‘ellipsoids＇ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type （1, 1） inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.
Bibliography:Heisenberg group, Hardy operator, Mp weight
11-5739/O1
In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on R, balls in Rn, or ‘ellipsoids＇ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type （1, 1） inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.
Hardy operator
Heisenberg group
Document received on :2015-10-13
M p weight
Document accepted on :2015-10-26
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ObjectType-Feature-1
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ObjectType-Article-1
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ISSN:1673-3452
2731-8648
1673-3576
2731-8656
DOI:10.1007/s11464-015-0508-5