Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T17:32:48.067Z Has data issue: false hasContentIssue false
  • English
  • Français

A PARABOLIC SINGULAR INTEGRAL OPERATOR WITH ROUGH KERNEL

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  01 April 2008

YANPING CHEN ,
YONG DING  and
DASHAN FAN
YANPING CHEN
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Applied Science School, University of Science and Technology, Beijing 100083, China (email: yanpingch@126.com)
YONG DING*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China (email: dingy@bnu.edu.cn)
DASHAN FAN
Affiliation:
Department of Mathematics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA Huazhong Normal University, Wuhan 430074, China (email: fan@uwm.edu)
*
✉For correspondence; e-mail: dingy@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.

Let Ω(y′) be an H1(Sn−1) function on the unit sphere satisfying a certain cancellation condition. We study the Lp boundedness of the singular integral operator where α≥n and ρ is a norm function which is homogeneous with respect to certain nonistropic dilation. The result in the paper substantially improves and extends some known results.

Keywords

parabolic singular integralHardy spacerough kernel

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 2 , April 2008 , pp. 163 - 179
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The research was supported by NSF of China (Grant: 19371046 and 10571015) and SRFDP of China (Grant: 20050027025).

References

[1]Calderón, A. and Zygmund, A., ‘On the existence of certain singular integrals’, Acta Math. 88 (1952), 85–139.CrossRefGoogle Scholar
[2]Calderón, A. and Zygmund, A., ‘Singular integral operators and differential equations’, Amer. J. Math. 79 (1957), 901–921.CrossRefGoogle Scholar
[3]Colzani, L., ‘Hardy spaces on sphere’, PhD Thesis, Washington University, St Louis, MO, 1982.Google Scholar
[4]Ding, Y., Xue, Q. and Yabuta, K., ‘Parabolic Littlewood–Paley g-function with rough kernels’, Acta Math. Sin. (Engl. Ser.) 24 (2008), to appear.Google Scholar
[5]Duoandikoetxea, J. and Rubio de Francia, J. L., ‘Maximal and singular integral operators via Fourier transform estimates’, Invent. Math. 84 (1986), 541–561.CrossRefGoogle Scholar
[6]Fabes, E. B. and Rivière, N. M., ‘Singular integrals with mixed homogeneity’, Studia Math. 27 (1966), 19–38.CrossRefGoogle Scholar
[7]Fan, D. and Pan, Y., ‘A singular integral operator with rough kernel’, Proc. Amer. Math. Soc. 125 (1997), 3695–3703.CrossRefGoogle Scholar
[8]Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28 (Princeton University Press, Princeton, NJ, 1982).Google Scholar
[9]Nagel, A., Riviere, N. M. and Wainger, S., ‘On Hilbert transforms along curves, II’, Amer. J. Math. 98 (1976), 395–403.CrossRefGoogle Scholar
[10]Ricci, F. and Weiss, G., ‘A characterization of H 1(Σn−1)’, in: Harmonic Analysis and Euclidean Spaces, Proceedings of Symposia in Pure Mathematics, 35 (American Mathematical Society, Providence, RI, 1979), pp. 289–294.CrossRefGoogle Scholar
[11]Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, NJ, 1971).Google Scholar