PM  >> Vol. 7 No. 1 (January 2017)

    关于双线性Hardy算子的两个端点弱型估计
    Two Weak Endpoint Estimates on Bilinear Hardy Operators

  • 全文下载: PDF(377KB) HTML   XML   PP.43-49   DOI: 10.12677/PM.2017.71007  
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作者:  

张春梅:安徽师范大学数学与计算机科学学院,安徽 芜湖

关键词:
Hardy算子双线性Hardy算子Lp空间Morrey空间Hardy Operator Multiple Hardy Operator Lp Space Morrey Space

摘要:

通过严格的计算,对双线性Hardy算子在中心Morrey空间和加权Lebesgue空间上的边界进行了端点情形下的弱型估计,这是对现有理论的有益补充。

By strict calculation, we mainly give the weak estimate for the boundary of the bilinear Hardy operator on the Morrey space and the weighted Lebesgue space, which is a useful supplement to the existing theory.

文章引用:
张春梅. 关于双线性Hardy算子的两个端点弱型估计[J]. 理论数学, 2017, 7(1): 43-49. http://dx.doi.org/10.12677/PM.2017.71007

参考文献

[1] Anderson, K. and Muckenhoupt, B. (1982) Weighted Weak Type Hardy Inequalities with Application to Hilbert Transforms and Maximal Functions. Studia Mathematica, 72, 9-26.
[2] Chen, J.C., Fan, D.S. and Wang, S.L. (2013) Hausdorff Operators on Euclidean Spaces. Applied Mathematics—A Journal of Chinese Universities, 28, 548-564. https://doi.org/10.1007/s11766-013-3228-1
[3] Bennet, C., Devore, R.A. and Sharpley, R.C. (1981) Weak and BMO. Annals of Mathematics, 113, 601-611. https://doi.org/10.2307/2006999
[4] Bliss, G.A. (1930) An Integral Inequality. Journal London Mathematical Society, 317, 40-46. https://doi.org/10.1112/jlms/s1-5.1.40
[5] Broadbent, T.A.A. (1928) A Proof of Hardy’s Convergence Theorem. Journal London Mathematical Society, 3, 242- 243. https://doi.org/10.1112/jlms/s1-3.4.242
[6] Levinson, N. (1964) Generalizations of an Inequality of Hardy. Duke Mathematical Journal, 31, 389-394. https://doi.org/10.1215/S0012-7094-64-03137-0
[7] Muckenhoupt, B. (1972) Hardy’s Inequality with Weights. Studia Mathematica, 44, 31-38.
[8] Golubov, B.I. (1997) Boundedness of the Hardy and the Hardy-Littlewood Operators in the Spaces ReH1 and BMO. Sbornik: Mathematics, 188, 1041-1054. https://doi.org/10.1070/SM1997v188n07ABEH000246
[9] Rakotondratsimba, Y. (1998) On the Boundedness of Classical Operators on Weighted Lorentz Spaces. Georgian Mathematical Journal, 5, 177-200. https://doi.org/10.1007/BF02767995
[10] Calderon, A.P. (1966) Space between and and the Theorem of Marcikiewiez. Studia Mathematica, 26, 273- 299.
[11] Hardy, G.H. (1928) Note on Some Points in the Integral Calculus. Messenger of Mathematics, 57, 12-16.
[12] Martin-Reyes, F. and Ortega, P. (1998) On Weighted Weak Type Inequalities for Modified Hardy Operators. Proceedings of the American Mathematical Society, 126, 1739-1746. https://doi.org/10.1090/S0002-9939-98-04247-6
[13] Long, R.L. (1985) Hp Martingale Theory. Peking University Press, Beijing.
[14] Xiao, J. (2001) and BMO Bounds of Weighted Hardy-Littlewood Averages. Journal of Mathematical Analysis and Applications, 262, 660-666. https://doi.org/10.1006/jmaa.2001.7594
[15] Hardy, G.H., Littlewood, J.E. and Polya, G. (1934) Inequalities. Cambridge University Press, London and New York.
[16] Edmunds, D., Gurka, P. and Pick, L. (1994) Compactness of Hardy Type Operators in Weighted Banach Function Spaces. Studia Mathematica, 109, 73-90.
[17] Christ, M. and Grafakos, L. (1995) Best Constants for Two Non-Convolution Inequalities. Proceedings of the American Mathematical Society, 123, 1687-1687. https://doi.org/10.1090/S0002-9939-1995-1239796-6
[18] Gao, G., Hu, X. and Zhang, C. (2016) Sharp Weak Estimates for Hardy-Type Operators. Annals of Functional Analysis, 7, 421-433. https://doi.org/10.1215/20088752-3605447
[19] Zhao, F.Y., Fu, Z.W. and Lu, S.Z. (2012) Endpoint Estimates for n-Dimensional Hardy Operators and Their Commutators. Science China Mathematics, 55, 1977-1990. https://doi.org/10.1007/s11425-012-4465-0
[20] Fu, Z., Grafakos, L., Lu, S.Z., et al. (2012) Sharp Bounds for m-Linear Hardy and Hilbert Operators. Houston Journal of Mathematics, 38, 225-244.
[21] Gao, G. and Zhao, F. (2015) Sharp Weak Bounds for Hausdorff Operators. Analysis Mathematica, 41, 163-173. https://doi.org/10.1007/s10476-015-0204-4