Lp Boundedness for Singular Integral Operators with L(log+ L)2 Kernels on Product Spaces

  • Received : 2005.03.03
  • Published : 2006.09.23

Abstract

In this paper, we study the $L^p$ mapping properties of singular integral operators related to homogeneous mappings on product spaces with kernels which belong to $L(log^+\;L)^2$. Our results extend as well as improve some known results on singular integrals.

Keywords

References

  1. A. Al-Salman, H. Al-Qassem and Y. Pan, Singular integrals associated to homogeneous mappings with rough kernels, Hokkaido Mathematical Journal, 33(2004), 551-569. https://doi.org/10.14492/hokmj/1285851910
  2. A. Al-Salman, H. Al-Qassem and Y. Pan, Singular Integrals on Product Domains, Indiana Univ. Math. J., 55(1)(2006), 369-387. https://doi.org/10.1512/iumj.2006.55.2626
  3. H. Al-Qassem and Y. Pan, $L^{p}$ boundedness for singular integrals with rough kernels on product domains, Hokkaido Math. J., 31(2002), 555-613. https://doi.org/10.14492/hokmj/1350911903
  4. A. Al-Salman and Y. Pan, Singular integrals with rough kernels in $Llog^{+}L(Sn^{n-1})$, J. London Math. Soc., 66(2)(2002), 153-174. https://doi.org/10.1112/S0024610702003241
  5. Calderon, A. P. and Zygmund, A., On singular integrals, Amer. J. Math., 78(1956), 289-309. https://doi.org/10.2307/2372517
  6. L. Cheng, Singular integrals related to homogeneous mappings, Michigan Math. J., 47(2)(2000), 407-416. https://doi.org/10.1307/mmj/1030132544
  7. J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36(1986), 185-206. https://doi.org/10.5802/aif.1073
  8. J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math., 84(1986), 541-561. https://doi.org/10.1007/BF01388746
  9. D. Fan, K. Guo and Y. Pan, Singular integrals with rough kernels on product spaces, Hokkaido Math. J., 28(1999), 435-460. https://doi.org/10.14492/hokmj/1351001230
  10. D. Fan, K. Guo and Y. Pan, $L^{p}$ estimates for singular integrals associated to homogeneous surfaces, J. Reine Angew. Math., 542(2002), 1-22.
  11. R. Fefferman, Singular integrals on product domains, Bull. Amer. Math. Soc., 4(1981), 195-201. https://doi.org/10.1090/S0273-0979-1981-14883-7
  12. R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45(1982), 117-143. https://doi.org/10.1016/S0001-8708(82)80001-7
  13. Y. Jiang and S. Lu, A class of singular integral operators with rough kernels on product domains, Hokkaido Math. J., 24(1995), 1-7. https://doi.org/10.14492/hokmj/1380892533
  14. F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, Jour. Func. Anal., 73(1987), 179-194. https://doi.org/10.1016/0022-1236(87)90064-4
  15. F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier, 42(1992), 637-670. https://doi.org/10.5802/aif.1304
  16. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970.
  17. E. M. Stein, Harmonic analysis real-variable methods, orthogonality and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.