Korean J. Math.  Vol 26, No 1 (2018)  pp.103-127
DOI: http://dx.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).

Full Text:

PDF

References


M. A. Arin ̃o 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. (Google Scholar)

A ́. B ́enyi and T. Oh, Best constants for certain multilinear integral operators. J. Inequal. Appl., 2006, 1–12. (Google Scholar)

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Academic Press, 1988. (Google Scholar)

S. Boza and J. Soria, Norm estimates for the Hardy operator in terms of Bp weights, Proc. Amer. Math. Soc 145 (2017), 2455–2465. (Google Scholar)

J. Chen, D. Fan and J. Li, Hausdorff operators on function spaces, Chin. Ann. Math. Ser. B 33 (2012), 537–556. (Google Scholar)

J. Chen, D. Fan, X. Lin and J. Ruan, The fractional Hausdorff operator on Hardy spaces, Anal. Math. 42 (2016), 1–17. (Google Scholar)

J. Chen, D. Fan and S. Wang, Hausdorff operators on Eulidean spaces, Appl. Math. J. Chinese Univ. Ser. B 28 (2013), 548–564. (Google Scholar)

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. (Google Scholar)

M. Carro, A. Garc ́ıa del Amo and J. Soria, Weak type weights and normable Lorentz spaces, Proc. Amer. Math. Soc 124 (1996), 849–857. (Google Scholar)

M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator J. Funct. Anal. 112 (1993), 480–494. (Google Scholar)

M. J. Carro and J. Soria, The Hardy-Littlewood maximal function and weighted Lorentz spaces, J. London Math. Soc. 55 (1997), 146–158. (Google Scholar)

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. (Google Scholar)

P. Dr ́abek, H. P. Heinig and A. Kufner, Higher dimensional Hardy inequality, Internat. Ser. Numer. Math. 123 (1997), 3–16. (Google Scholar)

Z. Fu, L. Grafakos, S. Lu, et al., Sharp bounds for m-linear Hardy and Hilbert operators, Houston J. Math. 38 (2012), 225–244. (Google Scholar)

D. Fan and F. Zhao, Multilinear fractional Hausdorff operators, Acta Math. Sin., Engl. Ser. 30 (2014), 1407–1421. (Google Scholar)

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 . (Google Scholar)

G. Gao and F. Zhao, Sharp weak bounds for a class of Hausdorff operator, Anal. Math. 41 (2015), 163–173. (Google Scholar)

R. A. Hunt, On L(p, q) spaces, Enseignement Math. 12 (1966), 249–276. (Google Scholar)

W. A. Hurwitz and L. L. Silverman, The consistency and equivalence of certain definitions of summabilities, Trans. Amer. Math. Soc. 18 (1917), 1–20. (Google Scholar)

A. Kamin ́ska, L. Maligranda, Order convexity and concavity of Lorentz spaces Λp,w, 0 < p < ∞, Studia Math. 160 (2004), 267–286. (Google Scholar)

A. Lerner and E. Liflyand, Multidimensional Hausdorff operators on the real Hardy spaces, J. Austral. Math. Soc. 83 (2007), 79–86. (Google Scholar)

H. Li and A. Kamin ́ska, Boundedness and compactness of Hardy operator on Lorentz-type spaces, Math. Nachr., DOI 10.1002/mana.201600049. (Google Scholar)

E. Liflyand, Hausdorff operators on Hardy Spaces, Eurasian Math. J. 4 (2013), 101–141. (Google Scholar)

E. Liflyand and A. Miyachi, Boundedness of the Hausdorff operators in Hp spaces, 0 < p < 1, Studia Math. 194 (2009), 279–292. (Google Scholar)

E. Liflyand and F. M ́oricz, The Hausdorff operator is bounded on real H1 space, Proc. Amer. Math. Soc. 128 (2000), 1391–1396. (Google Scholar)

S. Lu, D. Yan and F. Zhao, Sharp bounds for Hardy type operators on higher-dimensional product spaces, J. Inequal. Appl. 2013, 1–11. (Google Scholar)

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Group UK Limited, London, 1990. (Google Scholar)

J. Ruan and D. Fan, Hausdorff operators on the power weighted Hardy spaces, J. Math. Anal. Appl. 433 (2016), 31–48. (Google Scholar)

J. Ruan and D. Fan, Hausdorff operators on the weighted Herz-type Hardy spaces, Math. Inequal. Appl. 19 (2016), 565–587. (Google Scholar)

J. Ruan and D. Fan, Hausdorff type operators on the power weighted Hardy spaces, Math. Nachr., 2017, 00:1-14. https://doi.org/10.1002/mana.201600257. (Google Scholar)

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. (Google Scholar)

E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145–158. (Google Scholar)

J. Soria, Lorentz spaces of weak-type, Quart. J. Math. Oxford Ser. 49 (1998), 93–103. (Google Scholar)

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, 1993. (Google Scholar)

S. Thangavelu, Harmonic analysis on the Heisenberg group, Progr. Math., vol. 159, Birkh¡§auser, Boston, 1998. (Google Scholar)

X. Wu and J. Chen, Best constant for Hausdorff operators on n-dimensional product spaces, Sci. China Math. 57 (2014), 569–578. (Google Scholar)

Q. Wu and Z. Fu, Sharp estimates for the Hardy operator on the Heisenberg group, Front. Math. China 11 (2016), 155–172. (Google Scholar)

F. Zhao, Z. Fu and S. Lu, Endpoint estimates for n-dimensional Hardy operators and their commutators, Sci. China Math. 55 (2012), 1977–1990. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr