Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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HAUSDORFF OPERATORS ON WEIGHTED LORENTZ SPACES

  • Sun, Qinxiu (Department of Mathematics Zhejiang University of Science and Technology) ;
  • Fan, Dashan (Department of Mathematics University of Wisconsin-Milwaukee) ;
  • Li, Hongliang (Department of Mathematics Zhejiang International Studies University)
  • Received : 2018.01.06
  • Accepted : 2018.03.18
  • Published : 2018.03.30
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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 ${\Lambda}^p_u(w)$ and the weak-type weighted Lorentz spaces ${\Lambda}^{p,{\infty}}_u(w)$ are investigated. Especially, the results cover the classical power weighted space $L^{p,q}_{\alpha}$. The results are also extended to the product spaces ${\Lambda}^{p_1}_{u_1}(w_1){\times}{\Lambda}^{p_2}_{u_2}(w_2)$, especially for $L^{p_1,q_1}_{{\alpha}_1}{\times}L^{p_2,q_2}_{{\alpha}_2}$. 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

Acknowledgement

Supported by : National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province of China

References

  1. M. A. Arino and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735.
  2. A. Benyi and T. Oh, Best constants for certain multilinear integral operators. J. Inequal. Appl., 2006, 1-12.
  3. C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Academic Press, 1988.
  4. S. Boza and J. Soria, Norm estimates for the Hardy operator in terms of $B_p$ weights, Proc. Amer. Math. Soc 145 (2017), 2455-2465. https://doi.org/10.1090/proc/13604
  5. J. Chen, D. Fan and J. Li, Hausdorff operators on function spaces, Chin. Ann. Math. Ser. B 33 (2012), 537-556. https://doi.org/10.1007/s11401-012-0724-1
  6. J. Chen, D. Fan, X. Lin and J. Ruan, The fractional Hausdorff operator on Hardy spaces, Anal. Math. 42 (2016), 1-17. https://doi.org/10.1007/s10476-016-0101-5
  7. J. Chen, D. Fan and S. Wang, Hausdorff operators on Eulidean spaces, Appl. Math. J. Chinese Univ. Ser. B 28 (2013), 548-564. https://doi.org/10.1007/s11766-013-3228-1
  8. J. Chen, D. Fan and C. Zhang, Boundedness of Hausdorff operators on some product Hardy type spaces, Appl. Math. J. Chinese Univ. 27 (2012), 114-126. https://doi.org/10.1007/s11766-012-2922-8
  9. M. Carro, A. Garcia del Amo and J. Soria, Weak type weights and normable Lorentz spaces, Proc. Amer. Math. Soc 124 (1996), 849-857. https://doi.org/10.1090/S0002-9939-96-03214-5
  10. M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator J. Funct. Anal. 112 (1993), 480-494. https://doi.org/10.1006/jfan.1993.1042
  11. M. J. Carro and J. Soria, The Hardy-Littlewood maximal function and weighted Lorentz spaces, J. London Math. Soc. 55 (1997), 146-158. https://doi.org/10.1112/S0024610796004462
  12. M. J. Carro, J. A. Raposo and J. Soria, Recent developements in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187, 2007.
  13. P. Drabek, H. P. Heinig and A. Kufner, Higher dimensional Hardy inequality, Internat. Ser. Numer. Math. 123 (1997), 3-16.
  14. Z. Fu, L. Grafakos, S. Lu, et al., Sharp bounds for m-linear Hardy and Hilbert operators, Houston J. Math. 38 (2012), 225-244.
  15. D. Fan and F. Zhao, Multilinear fractional Hausdorff operators, Acta Math. Sin., Engl. Ser. 30 (2014), 1407-1421. https://doi.org/10.1007/s10114-014-3552-2
  16. J. H. Guo, L. J. Sun and F. Y. Zhao, Hausdorff Operators on the Heisenberg Group, Acta Math. Sin. (Engl. Ser.) 31 (2015), 1703-1714 . https://doi.org/10.1007/s10114-015-5109-4
  17. G. Gao and F. Zhao, Sharp weak bounds for a class of Hausdorff operator, Anal. Math. 41 (2015), 163-173. https://doi.org/10.1007/s10476-015-0204-4
  18. R. A. Hunt, On L(p, q) spaces, Enseignement Math. 12 (1966), 249-276.
  19. W. A. Hurwitz and L. L. Silverman, The consistency and equivalence of certain definitions of summabilities, Trans. Amer. Math. Soc. 18 (1917), 1-20. https://doi.org/10.1090/S0002-9947-1917-1501058-2
  20. A. Kaminska, L. Maligranda, Order convexity and concavity of Lorentz spaces $A_{k,w}$, 0 < p < TEX>${\infty}$, Studia Math. 160 (2004), 267-286. https://doi.org/10.4064/sm160-3-5
  21. A. Lerner and E. Li yand, Multidimensional Hausdorff operators on the real Hardy spaces, J. Austral. Math. Soc. 83 (2007), 79-86. https://doi.org/10.1017/S1446788700036399
  22. H. Li and A. Kaminska, Boundedness and compactness of Hardy operator on Lorentz-type spaces, Math. Nachr., DOI 10.1002/mana.201600049.
  23. E. Liflyand, Hausdorff operators on Hardy Spaces, Eurasian Math. J. 4 (2013), 101-141.
  24. E. Lifiyand and A. Miyachi, Boundedness of the Hausdorff operators in $H_p$ spaces, 0 < p < 1, Studia Math. 194 (2009), 279-292. https://doi.org/10.4064/sm194-3-4
  25. E. Li yand and F. Moricz, The Hausdorff operator is bounded on real $H_1$ space, Proc. Amer. Math. Soc. 128 (2000), 1391-1396. https://doi.org/10.1090/S0002-9939-99-05159-X
  26. S. Lu, D. Yan and F. Zhao, Sharp bounds for Hardy type operators on higher-dimensional product spaces, J. Inequal. Appl. 2013, 1-11.
  27. B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Group UK Limited, London, 1990.
  28. J. Ruan and D. Fan, Hausdorff operators on the power weighted Hardy spaces, J. Math. Anal. Appl. 433 (2016), 31-48. https://doi.org/10.1016/j.jmaa.2015.07.062
  29. J. Ruan and D. Fan, Hausdorff operators on the weighted Herz-type Hardy spaces, Math. Inequal. Appl. 19 (2016), 565-587.
  30. J. Ruan and D. Fan, Hausdorff type operators on the power weighted Hardy spaces $H^p_{|\ast|}^{\alpha}$ ($\mathbb{R}^n$), Math. Nachr., 2017, 00:1-14. https://doi.org/10.1002/mana.201600257.
  31. J. Ruan, D. Fan and Q. Wu, Weighted Herz space estimates for Hausdorff operators on the Heisenberg group, Banach J. Math. Appl. 11 (2017), 513-535. https://doi.org/10.1215/17358787-2017-0004
  32. E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158. https://doi.org/10.4064/sm-96-2-145-158
  33. J. Soria, Lorentz spaces of weak-type, Quart. J. Math. Oxford Ser. 49 (1998), 93-103. https://doi.org/10.1093/qmathj/49.1.93
  34. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, 1993.
  35. S. Thangavelu, Harmonic analysis on the Heisenberg group, Progr. Math., vol. 159, Birkhisxauser, Boston, 1998.
  36. X. Wu and J. Chen, Best constant for Hausdorff operators on n-dimensional product spaces, Sci. China Math. 57 (2014), 569-578. https://doi.org/10.1007/s11425-013-4725-7
  37. Q. Wu and Z. Fu, Sharp estimates for the Hardy operator on the Heisenberg group, Front. Math. China 11 (2016), 155-172. https://doi.org/10.1007/s11464-015-0508-5
  38. F. Zhao, Z. Fu and S. Lu, Endpoint estimates for n-dimensional Hardy operators and their commutators, Sci. China Math. 55 (2012), 1977-1990. https://doi.org/10.1007/s11425-012-4465-0