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

Qinxiu Sun, Dashan Fan, Hongliang Li

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.

Keywords


Hausdorff operators, weighted Lorentz spaces, sharp bounds.

Subject classification

46E30, 46B42

Sponsor(s)

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

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