Abstract
The authors mainly study the Hausdorff operators on Euclidean space ℝn. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces \(H\dot K_q^{\alpha ,p} (\mathbb{R}^n )\) than their performance on the Hardy spaces H p(ℝn) when 0 < p < 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.
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Carro, M. and Soria, J., Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal., 112, 1993, 480–494.
Chen, J. C., Fan, D. S. and Zhang, C. J., Boundedness of Hausdorff operators on some product Hardy type spaces, Appl. Math. Jour. Chin. Univ. (Ser. B), 27(1), 2012, 114–126.
Chen, J. C., Fan, D. S. and Zhang, C. J., Multilinear Hausdorff operators and their best constants, Acta Math. Sinica (Ser. B), to appear.
Christ, M. and Grafakos, L., The best constants for two non-convolution inequalities, Proc. Amer. Math. Soc., 123, 1995, 1687–1693.
Frasier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93, 1990, 34–170.
Frasier, M., Jawerth, B. and Weiss, G., Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series on Math., 79, A. M. S., Providence, RI, 1991.
Fu, Z., Liu, Z., Lu, S., et al., Characterization for commutators of n-dimensional fractional Hardy operators, Sci. China, Ser. A, 50, 2007, 1418–1426.
Han, Y., Paluszynski, M. and Weiss, G., A new atomic decomposition for the Triebel-Lizorkin spaces, Harmonic Analysis and Operator Theory, Caracas, 1994, 235–249, Contemp. Math., 189, A. M. S., Providence, RI, 1995.
Lerner, A. K. and Liflyand, E., Multidimensional Hausdorff operators on the real Hardy spaces, J. Aust. Math. Soc., 83, 2007, 79–86.
Liflyand, E., Open problems on Hausdorff operators, Complex Analysis and Potential Theory, Proceedings of the Conference Satellite to ICM 2006, Istanbul, Turkey, 2006, 280–284.
Liflyand, E. and Miyachi, A., Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1, Studia Math., 194, 2009, 279–292.
Liflyand, E. and Mórecz, F., The Hausdorff operator is bounded on real H 1 space, Proc. Amer. Math. Soc., 128, 2000, 1391–1396.
Lu, S., Four Lectures on Real Hardy Spaces, World Scientific Press, Beijing, 1995.
Lu, S., Yang, D. and Hu, G., Herz Type Spaces and Their Applications, Science Press, Beijing, 2008.
Torres, R., Boundedness Results for Operators with Singular Kernels on Distribution Spaces, Mem. Amer. Math. Soc., 90(442), Providence, RI, 1991.
Triebel, H., Theory of Function Spaces, Birkhäuser, Basel, Boston, Stuttgart, 1983.
Zhong, Y., Estimates on Certain Operators in Function Spaces, Ph.D. Thesis, 2011, Zhejiang University, Hangzhou, China.
Zhu, X. R. and Chen, J. C., Integrability of the general product Hardy operators on the product Hardy spaces, Appl. Math. Jour. Chin. Univ. (Ser. B), 27(2), 2012, 225–233.
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Project supported by the National Natural Science Foundation of China (Nos. 10931001, 10871173).
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Chen, J., Fan, D. & Li, J. Hausdorff operators on function spaces. Chin. Ann. Math. Ser. B 33, 537–556 (2012). https://doi.org/10.1007/s11401-012-0724-1
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DOI: https://doi.org/10.1007/s11401-012-0724-1