Weighted Hardy operators and commutators on Morrey spaces

Abstract

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝⁿ)) on Morrey spaces.

Keywords

Weighted Hardy operator / BMO / commutator / Morrey space

References

[1] Anderson K. F., Muckenhoupt B. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 1982, 72: 9-26.

[2] Bastero J., Milman M., Ruiz F. J. Commutators of the maximal and sharp functions. Proc Amer Math Soc, 2000, 128: 3329-3334.

[3] Carton-Lebrun C., Fosset M. Moyennes et quotients de Taylor dans BMO. Bull Soc Roy Sci Liège, 1984, 53: 85-87.

[4] Chiarenza F., Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Mat, 1987, 7: 273-279.

[5] Coifman R. R., Rochberg R., Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103: 611-635.

[6] Ding Y., Lu S. Z., Yabuta K. On commutators of Marcinkiewicz integrals with rough kernel. J Math Anal Appl, 2002, 275: 60-68.

[7] Duong X. T., Yan L. X. On commutators of fractional integrals. Proc Amer Math Soc, 2002, 132: 3549-3557.

[8] Fu Z. W., Liu Z. G., Lu S. Z. Commutators of weighted Hardy operators. Proc Amer Math Soc, 2009, 137: 3319-3328.

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

[10] Hardy G. H., Littlewood J., Pólya G. Inequalities, 1952 2nd ed. London/New York: Cambridge University Press.

[11] Komori Y. Notes on commutators of Hardy operators. Int J Pure Appl Math, 2003, 7: 329-334.

[12] Lacey M. T. Commutators with Riesz potentials in one and several parameters. Hokkaido Math J, 2007, 36: 175-191.

[13] Long S. C., Wang J. Commutators of Hardy operators. J Math Anal Appl, 2002, 274: 626-644.

[14] Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126-166.

[15] Muckenhoupt B. Hardy’s inequality with weights. Studia Math, 1972, 44: 31-38.

[16] Perez C. Endpoints for commutators of Singular integrals operators. J Funct Anal, 1995, 128: 163-185.

[17] Xiao J. L^p and BMO bounds of weighted Hardy-Littlewood averages. J Math Anal Appl, 2001, 262: 660-666.