Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T19:54:11.160Z Has data issue: false hasContentIssue false
  • English
  • Français

The higher order commutators of the fractional integrals on Hardy spaces

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

Shunchao Long  and
Jian Wang
Shunchao Long
Affiliation:
Mathematics DepartmentXiangtan University Xiangtan, 411105 P. R.China e-mail: sclong@xtu.edu.cn, jwang@hnedu.com
Jian Wang
Affiliation:
Mathematics DepartmentXiangtan University Xiangtan, 411105 P. R.China e-mail: sclong@xtu.edu.cn, jwang@hnedu.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate the boundedness on Hardy spaces for the higher order commutator Tb, m generated by the BMO function b and fractional integral type operator Tτ, and establish the boundness theorems for Tτb, m from Hp1.q1.sb, m to Lp2 and to Hp2 (0 < p1 ≤ 1), and from H Ka. p1.sq1, b, m to Ka.p2q2 and to H Ka. p2q2, respectively, for certain ranges of α, p1, q1, p2, q2 and s.

Keywords

Hardy spaceHerz spacecommutatorfractional integral type operatorBMO.

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 1 , August 2005 , pp. 11 - 24
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Alvarez, J., ‘Contiunity properties for linear commutators of Calderón-Zygmund operators’, Collect. Math. 48 (1998), 17–31.Google Scholar
[2]Alvarez, J., Bagby, R. J., Kurtz, D. S. and Perez, C., ‘Weighted estimates for commutators of linear operators’, Studia Math. 104 (1993), 195–209.Google Scholar
[3]Bloom, S., ‘A commutator theorem and weighted BMO’, Trans. Amer. Math. Soc. 292 (1985), 103–122.CrossRefGoogle Scholar
[4]Chanillo, S., ‘A note on commutators’, Indiana Univ. Math. J. 31 (1982), 7–16.Google Scholar
[5]Chen, Y. and Lau, K., ‘On some new classes of Hardy spaces’, J. Funct. Anal. 84 (1989), 225–278.CrossRefGoogle Scholar
[6]Coifman, R. R., Rochberg, R. and Weiss, G., ‘Factorization theorems for Hardy spaces in several variables’, Ann. of Math. (2) 103 (1976), 611–635.Google Scholar
[7]Coifman, R. R. and Weiss, G., ‘Extensions of Hardy spaces and their use in analysis’, Bull. Amer. Math. Soc. 83 (1977), 569–645.Google Scholar
[8]Ding, Y. and Lu, S., ‘Higher order commutators for a class of rough operators’, Ark. Mat. 37 (1999), 33–44.Google Scholar
[9]Garcia-Cuerva, J., ‘Hardy spaces and Beurling algebras’, J. London Math. Soc. 39 (1989), 499–513.Google Scholar
[10]Garcia-Cuerva, J., Harboure, E., Segovia, C. and Torrea, J., ‘Weighted norm inequalities for commutators of strongly singular integrals’, Indiana Univ. Math. J. 40 (1991), 1397–1420.Google Scholar
[11]Grafakos, L., Li, X. and Yang, D., ‘Bilinear operators on Herz-type spaces’, Trans. Amer. Math. Soc. 350 (1998), 1249–1275.Google Scholar
[12]Hu, G., Lu, S. and Yang, D., ‘Boundedness of rough singular integral operators on homogeous Herz spaces’, J. Austral. Math. Soc. (Ser. A) 66 (1999), 201–233.Google Scholar
[13]Latter, R. H., ‘A decomposition of Hp(Rn) in terms of atoms’, Studia Math. 62, (1977), 92–101.Google Scholar
[14]Li, X. and Yang, D., ‘Boundedness of some sublinear operators on Herz spaces’, Illinois J. Math. 40 (1996), 484–501.Google Scholar
[15]Long, S. and Wang, J., ‘The higher order commutator theorem on Hardy spaces’, preprint, 1999.Google Scholar
[16]Lu, S. and Yang, D., ‘The local versions of Hp (Rn) spaces at the origin’, Studia Math. 116 (1995), 103–131.Google Scholar
[17]Lu, S. and Yang, D., ‘Weighted Hardy spaces associated with Herz spaces and its applications’, Sci. China (Ser. A) 3 (1995), 235–245 in Chinese.Google Scholar
[18]Lu, S. and Yang, D., ‘Hardy-Littlewood-Sobolev theorem of fractional integration on Herz-type spaces and its applications’, Canad. J. Math. 48 (1996), 363–380.Google Scholar
[19]Lu, S. and Yang, D., ‘The continuity of commutators on Herz-type spaces’, Michigan Math. J. 44 (1997), 255–281.CrossRefGoogle Scholar
[20]Paluszynski, M., Characterization of Lipschitz spaces via commutator of Coifman, Rochberg and Weiss: a multiplier theorem for the semigruop of contractions (Ph. D. Thesis, Washington University, St. Louis, 1992).Google Scholar
[21]Perez, C., ‘Endpoint estimates for commutators of singular integral operators’, J. Funct. Anal. 128 (1995), 163–185.CrossRefGoogle Scholar
[22]Segovia, C. and Torrea, J. L., ‘Vector-valued commutators and applications’, Indiana Univ. Math. J. 38 (1989), 959–971.Google Scholar
[23]Segovia, C., ‘Higher order commutators for vector-valued Calderón-Zygmund operators’, Trans. Amer. Math. Soc. 336 (1993), 537–556.Google Scholar
[24]Stein, E. M., Singular integrals and differentiability propertites of functions (Princeton Univ. Press, Princeton, NJ, 1970).Google Scholar
[25]Taibleson, M. and Weiss, G., ‘The molecular characterization of certain Hardy spaces’, Asterisque 77 (1980), 67–151.Google Scholar