Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-20T22:50:46.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundedness of rough singular integral operators on homogeneous Herz spaces

Part of: General theory of linear operators Harmonic analysis in several variables Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Guoen Hu ,
Shanzhen Lu  and
Dachun Yang
Guoen Hu
Affiliation:
Department of Applied Mathematics, Institute of Information Engineering, Box 1001-47, Zhengzhou 450002, P. R. China
Shanzhen Lu
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China e-mail: lusz@bnu.edu.cn e-mail: dcyang@bnu.edu.cn
Dachun Yang
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China e-mail: lusz@bnu.edu.cn e-mail: dcyang@bnu.edu.cn
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.

The authors establish the boundedness on the Herz spaces and the weak Herz spaces for a large class of rough singular integral operators and their corresponding fractional versions. Applications are given to Fefferman's rough singular integral operators, their fractional versions, their commutators with BMO() functions and Ricci-Stein oscillatory singular integral operators. Some new results are obtained.

Keywords

Herz spaceLebesgue spacepower weightsingular integralfractional singular integral

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 47A30: Norms (inequalities, more than one norm, etc.) 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 2 , April 1999 , pp. 201 - 223
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Chanillo, S. and Christ, M., ‘Weak (1, 1) bounds for oscillatory singular integrals’, Duke Math. J. 55 (1987), 141–155.CrossRefGoogle Scholar
[2]Christ, M. and Rubio de Francia, J. L., ‘Weak type (1, 1) bounds for rough operators, II’, Invent. Math. 93 (1988), 225–237.CrossRefGoogle Scholar
[3]Coifman, R. R. and Weiss, G., ‘Extensions of Hardy spaces and their use in analysis’, Bull. Amer. Math. Soc. 83 (1977), 569–645.CrossRefGoogle Scholar
[4]Ding, Y., Some problems on oscillatory integral operators and fractional integral operators with rough kernels (Ph.D. Thesis, Beijing Normal University, Beijing, 1995).Google Scholar
[5]Duoandikoetxea, J., ‘Weighted norm inequalities for homogeneous singular integrals’, Trans. Amer. Math. Soc. 336 (1993), 869–880.CrossRefGoogle Scholar
[6]Fan, D. and Pan, Y., ‘Singular integral operatiors with rough kernels supported by subvarieties’, Amer. J. Math. 119 (1997), 799–839.CrossRefGoogle Scholar
[7]Fefferman, R., ‘A note on singular integrals’, Proc. Amer. Math. Soc. 74 (1979), 266–270.CrossRefGoogle Scholar
[8]Hernández, E., and Yang, D., ‘Interpolation of Herz spaces and applications’, Math. Nachr. (to appear).Google Scholar
[9]Herz, C., ‘Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms’, J. Math. Mech. 18 (1967), 283–324.Google Scholar
[10]Hu, G., ‘A note on sublinear operators and power weights’, J. Beijing Normal Univ. (Natur. Sci.) 31 (1995), 170–174.Google Scholar
[11]Hu, G., ‘Weighted norm inequalities for commutators of homogeneous singular integrals’, Acta Math. Sinica (New Ser.) (Special Issue) 11 (1995), 77–88.Google Scholar
[12]Hu, G., Lu, S. and Yang, D., ‘The weak Herz spaces’, J. Beijing Normal Univ. (Natur. Sci.) 33 (1997), 27–34.Google Scholar
[13]Jiang, Y. and Lu, S., ‘Oscillatory singular integrals with rough kernel’, in: Harmonic Analysis in China (eds. Cheng, M., Deng, D., Gong, S. and Yang, C.), Mathematics and its Applications, vol. 327 (Kluwer Acad. Publ., London, 1995) pp. 135–145.CrossRefGoogle Scholar
[14]Li, X. and Yang, D., ‘Boundedness of some sublinear operators on Herz spaces’, Illinois J. Math. 40 (1996), 484–501.CrossRefGoogle Scholar
[15]Liu, W. and Lu, S., ‘Rough operators on the weighted Herz spaces’, J. Beijing Normal Univ. (Natur. Sci.) 31 (1995), 446–451.Google Scholar
[16]Lu, S., ‘An extension of Soria-Weiss' theorem on sublinear operators and power weights’, J. Beijing Normal Univ. (Natur. Sci.) 30 (1994), 313–316.Google Scholar
[17]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.CrossRefGoogle Scholar
[18]Muckenhoupt, B. and Wheeden, R. L., ‘Weighted norm inequalities for singular and fractional integrals’, Trans. Amer. Math. Soc. 161 (1971), 249–258.CrossRefGoogle Scholar
[19]Pan, Y., ‘Uniform weak (1, 1) bounds for oscillatory singular integrals’, in: Harmonic Analysis in China (eds. Cheng, M., Deng, D., Gong, S. and Yang, C.), Mathematics and its Applications, vol. 327 (Kluwer Acad. Publ., London, 1995) pp. 210–219.CrossRefGoogle Scholar
[20]Ricci, F. and Stein, E. M., ‘Harmonic analysis on nilpotent groups and singular integral, I’, J. Funct. Anal. 73 (1987), 179–194.CrossRefGoogle Scholar
[21]Seeger, A., ‘Singular integral operators with rough convolution kernels’, J. Amer. Math. Soc. 9 (1996), 95–105.CrossRefGoogle Scholar
[22]Sona, F. and Wesis, G., ‘A remark on singular integrals and power weights’, Indiana Univ. Math. J. 43 (1994), 187–204.Google Scholar
[23]Stein, E. M., ‘Note on singula integrals’, Proc. Amer. Math. Soc. 8 (1957), 250–254.CrossRefGoogle Scholar