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A PARABOLIC SINGULAR INTEGRAL OPERATOR WITH ROUGH KERNEL

  • YANPING CHEN (a1) (a2), YONG DING (a3) and DASHAN FAN (a4) (a5)

Abstract

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 αn 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.

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Copyright

Corresponding author

For correspondence; e-mail: dingy@bnu.edu.cn

Footnotes

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

Footnotes

References

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[1]Calderón, A. and Zygmund, A., ‘On the existence of certain singular integrals’, Acta Math. 88 (1952), 85139.
[2]Calderón, A. and Zygmund, A., ‘Singular integral operators and differential equations’, Amer. J. Math. 79 (1957), 901921.
[3]Colzani, L., ‘Hardy spaces on sphere’, PhD Thesis, Washington University, St Louis, MO, 1982.
[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.
[5]Duoandikoetxea, J. and Rubio de Francia, J. L., ‘Maximal and singular integral operators via Fourier transform estimates’, Invent. Math. 84 (1986), 541561.
[6]Fabes, E. B. and Rivière, N. M., ‘Singular integrals with mixed homogeneity’, Studia Math. 27 (1966), 1938.
[7]Fan, D. and Pan, Y., ‘A singular integral operator with rough kernel’, Proc. Amer. Math. Soc. 125 (1997), 36953703.
[8]Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28 (Princeton University Press, Princeton, NJ, 1982).
[9]Nagel, A., Riviere, N. M. and Wainger, S., ‘On Hilbert transforms along curves, II’, Amer. J. Math. 98 (1976), 395403.
[10]Ricci, F. and Weiss, G., ‘A characterization of H 1n−1)’, in: Harmonic Analysis and Euclidean Spaces, Proceedings of Symposia in Pure Mathematics, 35 (American Mathematical Society, Providence, RI, 1979), pp. 289294.
[11]Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, NJ, 1971).
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Keywords

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A PARABOLIC SINGULAR INTEGRAL OPERATOR WITH ROUGH KERNEL

  • YANPING CHEN (a1) (a2), YONG DING (a3) and DASHAN FAN (a4) (a5)

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