A Class of Singular Integral Operators Associated to Surfaces of Revolution

Muliang Cao, Huoxiong Wu

.

DOI: 10.4236/apm.2011.16060                

Abstract

In this paper, the authors establish the Lp-mapping properties of a class of singular integral operators along surfaces of revolution with rough kernels. The size condition on the kernels is optimal and much weaker than that for the classical Calderon-Zygmund singular integral operators.

Keywords

Singular Integrals, Surface of Revolution, Maximal Operator, Rough Kernel

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Al-Salman and Y. Pan, “Singular Integrals with Rough Kernels in L(log+L)1/2(Sn–1),” Journal of the London Mathematical Society, Vol. 66, No. 2, 2002, pp. 153-174. doi:10.1112/S0024610702003241
[2]	H. Al-Qassem and Y. Pan, “Singular Integrals along Surfaces of Revolution with Rough Kernels,” SUT Journal of Mathematics, Vol. 39, No. 1, 2003, pp. 55-70.
[3]	H. Carlsson, M. Christ, A. Cordoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger and D. Weinberg, “Lp Estimates for Maximal Functions and Hilbert Transforms along Flat Convex Plane Curves in R2,” Bulletin of the American Mathematical Society, Vol. 14, No. 2, 1986, pp. 263-267. doi:10.1090/S0273-0979-1986-15433-9
[4]	A. Carbery, M. Christ, J. Vance, S. Wainger and D. K. Watson, “Operators Associated to Flat Plane Curves: Lp Estimates via Dilation Methods,” Duke Mathematical Journal, Vol. 59, 1989, pp. 675-700. doi:10.1215/S0012-7094-89-05930-9
[5]	L. Chen and D. Fan, “On Singular Integrals along Surfaces Related to Block Spaces,” Integral Equations and Operator Theory, Vol. 29, 1997, pp. 261-268. doi:10.1007/BF01320700
[6]	L. C. Cheng and Y. Pan, “Lp Bounds For singular Integrals Associated to Surfaces of Revolution,” Journal of Mathematical Analysis and Applications, Vol. 265, 2002, pp. 163-169. doi:10.1006/jmaa.2001.7710
[7]	J. Duoandikoetxea and J. L. Rubio de Francia, “Maximal and Singular Integral Operators via Fourier Transform Estimates,” Inventiones Mathematicae, Vol. 84, No. 3, 1986, pp. 541-561. doi:10.1007/BF01388746
[8]	D. Fan and Y. Pan, “Singular Integral Operators with Rough Kernels Supported by Subvarieties,” American Journal of Mathematics, Vol. 119, No. 4, 1997, pp. 799- 839. doi:10.1353/ajm.1997.0024
[9]	D. Fan and Q. Zheng, “Maximal Singular Integral Operators along Surfaces,” Journal of Mathematical Analysis and Applications, Vol. 267, No. 2, 2002, pp. 746-759. doi:10.1006/jmaa.2001.7812
[10]	W. Kim, S. Wainger, J. Wright and S. Ziesler, “Singular Integrals and Maximal Functions Associated to Surfaces of Revolution,” Bulletin London Mathematical Society, Vol. 28, No. 3, 1996, pp. 291-296. doi:10.1112/blms/28.3.291
[11]	S. Lu, Y. Pan and D. Yang, “Rough Singular Integrals Associated to Surfaces of Revolution,” Proceedings of the AMS—American Mathematical Society, Vol. 129, 2001, pp. 2931-2940. doi:10.1090/S0002-9939-01-05893-2
[12]	Y. Pan, L. Tang and D. Yang, “Boundedness of Rough Singular Integrals Associated with Surfaces of Revolution,” Advances in Mathematics (China), Vol. 32, No. 2, 2003, pp. 677-682.
[13]	F. Ricci and E. M. Stein, “Multiparameter Singular Integrals and Maximal Functions,” Annales de l'institut Fourier, Vol. 42, No. 3, 1992, pp. 637-670. doi:10.5802/aif.1304
[14]	E. M. Stein, “Problem in Harmonic Analysis Related to Curvature and Oscillatory Integrals,” Proceedings of the International Conference on Mathematics, Berkely, 1986, pp. 196-221.
[15]	E. M. Stein and S. Wainger, “Problems in Harmonic Analysis Related to Curvature,” Bulletin of the American Mathematical Society, Vol. 84, No. 6, 1978, pp. 1239- 1295.
[16]	E. M. Stein, “Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals,” Princeton University Press, Princeton, 1993. doi:10.1090/S0002-9904-1978-14554-6
[17]	A. Al-Salman, “On Maximal Functions with Rough Kernels in L(log+L)1/2(Sn–1),” Collectanea Mathematica, Vol. 56, No. 1, 2005, pp. 47-56.
[18]	E. M. Stein, “Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, Princeton, 1970.