Korean J. Math. Vol. 26 No. 1 (2018) pp.103-127
DOI: https://doi.org/10.11568/kjm.2018.26.1.103

Hausdorff operators on weighted Lorentz spaces

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Published: Mar 30, 2018
Keywords:
Hausdorff operators, weighted Lorentz spaces, sharp bounds.
Mathematics Subject Classification:
46E30, 46B42

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Qinxiu Sun
Department of Mathematics Zhejiang University of Science and Technology Hangzhou, 310023, China
Dashan Fan
Department of Mathematics University of Wisconsin-Milwaukee Milwaukee, WI 53201, USA
Hongliang Li
Department of Mathematics Zhejiang International Studies University Hangzhou, 310012, China

Abstract

This paper is dedicated to studying some Hausdorff operators on the Heisenberg group $\mathbb{H}^{n}$. The sharp bounds on the strong-type weighted Lorentz spaces $\Lambda _{u}^{p}(w)$ and the weak-type weighted Lorentz spaces $ \Lambda _{u}^{p,\infty }(w)$ are investigated. Especially, the results cover the classical power weighted space $L_{\alpha}^{p,q}$. The results are also extended to the product spaces $\Lambda _{u_{1}}^{p_{1}}(w_{1})\times \Lambda_{u_{2}}^{p_{2}}(w_{2})$, especially for $L_{\alpha_{1}}^{p_{1},q_{1}}\times L_{\alpha _{2}}^{p_{2},q_{2}}$. Our proofs are quite different from those in previous documents since the duality principle, and some well-known inequalities concerning the weights are adopted. The results recover the existing results as well as we obtain new results in the new and old settings.


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Supporting Agencies

National Natural Science Foundation of China (11401530 11461033) NaturalScience Foundation of Zhejiang Province of China (LQ13A010018).

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