Hausdorff operators on function spaces

Abstract

The authors mainly study the Hausdorff operators on Euclidean space ℝ ⁿ. 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(ℝ ⁿ) 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.

Keywords

Hausdorff operator / Hardy operator / Hardy space / Herz space

References

[1] Carro M., Soria J.. Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal., 1993, 112: 480-494

[2] Chen J. C., Fan D. S., Zhang C. J.. Boundedness of Hausdorff operators on some product Hardy type spaces. Appl. Math. Jour. Chin. Univ. (Ser. B), 2012, 27(1): 114-126

[3] Chen, J. C., Fan, D. S. and Zhang, C. J., Multilinear Hausdorff operators and their best constants, Acta Math. Sinica (Ser. B), to appear.

[4] Christ M., Grafakos L.. The best constants for two non-convolution inequalities. Proc. Amer. Math. Soc., 1995, 123: 1687-1693

[5] Frasier M., Jawerth B.. A discrete transform and decompositions of distribution spaces. J. Funct. Anal., 1990, 93: 34-170

[6] Frasier M., Jawerth B., Weiss G.. Littlewood-Paley theory and the study of function spaces, 1991, Providence, RI: A. M. S.

[7] Fu Z., Liu Z., Lu S. Characterization for commutators of n-dimensional fractional Hardy operators. Sci. China, Ser. A, 2007, 50: 1418-1426

[8] Han Y., Paluszynski M., Weiss G.. A new atomic decomposition for the Triebel-Lizorkin spaces, Harmonic Analysis and Operator Theory, Caracas, 1994, 235–249, 1995, Providence, RI: A. M. S.

[9] Lerner A. K., Liflyand E.. Multidimensional Hausdorff operators on the real Hardy spaces. J. Aust. Math. Soc., 2007, 83: 79-86

[10] 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.

[11] Liflyand E., Miyachi A.. Boundedness of the Hausdorff operators in H ^p spaces, 0 < p < 1. Studia Math., 2009, 194: 279-292

[12] Liflyand E., Mórecz F.. The Hausdorff operator is bounded on real H ¹ space. Proc. Amer. Math. Soc., 2000, 128: 1391-1396

[13] Lu S.. Four Lectures on Real Hardy Spaces, 1995, Beijing: World Scientific Press

[14] Lu S., Yang D., Hu G.. Herz Type Spaces and Their Applications, 2008, Beijing: Science Press

[15] Torres, R., Boundedness Results for Operators with Singular Kernels on Distribution Spaces, Mem. Amer. Math. Soc., 90(442), Providence, RI, 1991.

[16] Triebel H.. Theory of Function Spaces, 1983, Basel, Boston, Stuttgart: Birkhäuser

[17] Zhong Y.. Estimates on Certain Operators in Function Spaces, 2011, Hangzhou, China: Zhejiang University

[18] Zhu X. R., Chen J. C.. Integrability of the general product Hardy operators on the product Hardy spaces. Appl. Math. Jour. Chin. Univ. (Ser. B), 2012, 27(2): 225-233