Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PARABOLIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO POLYNOMIALS COMPOUND CURVES AND EXTRAPOLATION

  • Liu, Feng (College of Mathematics and Systems Science Shandong University of Science and Technology) ;
  • Zhang, Daiqing (School of Mathematical Sciences Xiamen University)
  • Received : 2013.12.20
  • Published : 2015.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this note we consider the parametric Marcinkiewicz integrals with mixed homogeneity along polynomials compound curves. Under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the $L^p$ bounds of such operators are given by an extrapolation argument. Some previous results are greatly extended and improved.

Keywords

References

  1. H. Al-Qassem, L. C. Cheng, and Y. Pan, On certain rough integral operators and extrapolation, J. Inequal. Pure Appl. Math. 10 (2009), no. 3, Article 78, 15 pp.
  2. H. Al-Qassem and Y. Pan, On certain estimates for Marcinkiewicz integrals and extrapolation, Collect. Math. 60 (2009), no. 2, 123-145. https://doi.org/10.1007/BF03191206
  3. A. Al-Salman, A note on parabolic Marcinkiewicz integrals along surfaces, Proc. A. Razmadze Math. Inst. 154 (2010), 21-36.
  4. A. Al-Salman, On the $L^2$ boundedness of parametric Marcinkiewicz integral operator, J. Math. Anal. Appl. 375 (2011), no. 2, 745-752. https://doi.org/10.1016/j.jmaa.2010.09.062
  5. A. Al-Salman, H. Al-Qassem, L. C. Cheng, and Y. Pan, $L^p$ bounds for the function of Marcinkiewicz, Math. Res. Lett. 9 (2002), no. 5-6, 697-700. https://doi.org/10.4310/MRL.2002.v9.n5.a11
  6. A. P. Calderon and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. in Math. 16 (1975), 1-64. https://doi.org/10.1016/0001-8708(75)90099-7
  7. Y. Chen and Y. Ding, $L^p$ bounds for the parabolic Marcinkiewicz integral with rough kernels, J. Korean Math. Soc. 44 (2007), no. 3, 733-745. https://doi.org/10.4134/JKMS.2007.44.3.733
  8. Y. Chen and Y. Ding, The parabolic Littlewood-Paley operator with Hardy space kernels, Canad. Math. Bull. 52 (2009), no. 4, 521-534. https://doi.org/10.4153/CMB-2009-053-8
  9. Y. Ding, D. Fan, and Y. Pan, $L^p$-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 4, 593-600. https://doi.org/10.1007/s101140000015
  10. Y. Ding, D. Fan, and Y. Pan, On the $L^p$ boundedness of Marcinkiewicz integrals, Michigan Math. J. 50 (2002), no. 1, 17-26. https://doi.org/10.1307/mmj/1022636747
  11. Y. Ding, S. Lu, and K. Yabuta, A problem on rough parametric Marcinkiewicz functions, J. Aust. Math. Soc. 72 (2002), no. 1, 13-21. https://doi.org/10.1017/S1446788700003542
  12. Y. Ding, Q. Xue, and K. Yabuta, A remark to the $L^2$ boundedness of parametric Marcinkiewicz integral, J. Math. Anal. Appl. 387 (2012), no. 2, 691-697.
  13. Y. Ding, Q. Xue, and K. Yabuta, Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces, Tohoku Math. J. 62 (2010), no. 2, 233-262. https://doi.org/10.2748/tmj/1277298647
  14. J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541-561. https://doi.org/10.1007/BF01388746
  15. E. Fabes and N. Riviere, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19-38.
  16. D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997), no. 4, 799-839. https://doi.org/10.1353/ajm.1997.0024
  17. L. Hormander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-104. https://doi.org/10.1007/BF02547187
  18. F. Liu, On singular integrals associated to surfaces, Tohoku Math. J. 66 (2014), no. 1, 1-14. https://doi.org/10.2748/tmj/1396875659
  19. F. Liu and H. Wu, Rough Marcinkiewicz integrals with mixed homogeneity on product spaces, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 7, 1231-1244. https://doi.org/10.1007/s10114-013-1675-5
  20. W. R. Madych, On Littlewood-Paley functions, Studia Math. 50 (1974), 43-63.
  21. M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions, Studia Math. 135 (1999), no. 2, 103-142.
  22. S. Sato, Estimates for singular integrals and extrapolation, Studia Math. 192 (2009), no. 3, 219-233. https://doi.org/10.4064/sm192-3-2
  23. E. M. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. https://doi.org/10.1090/S0002-9947-1958-0112932-2
  24. T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203-217.
  25. F. Wang, Y. Chen, and W. Yu, $L^p$ bounds for the parabolic Littlewood-Paley operator associated to surfaces of revolution, Bull. Korean Math. Soc. 49 (2012), no. 4, 787-797. https://doi.org/10.4134/BKMS.2012.49.4.787
  26. H. Wu, On Marcinkiewicz integral operator with rough kernels, Integral Equations Operator Theory 52 (2005), no. 2, 285-298. https://doi.org/10.1007/s00020-004-1339-z
  27. H. Wu, $L^p$ bounds for Marcinkiewicz integrals associates to surfaces of revolution, J. Math. Anal. Appl. 321 (2006), no. 2, 811-827. https://doi.org/10.1016/j.jmaa.2005.08.087
  28. Q. Xue, Y. Ding, and K. Yabuta, Parabolic Littlewood-Paley g-function with rough kernel, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 12, 2049-2060. https://doi.org/10.1007/s10114-008-6338-6

Cited by

  1. $$L^p$$ L p bounds for Marcinkiewicz integrals associated to homogeneous mappings vol.181, pp.4, 2016, https://doi.org/10.1007/s00605-016-0968-z
  2. BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES vol.99, pp.03, 2015, https://doi.org/10.1017/S1446788715000191
  3. Two Quantum Coins Sharing a Walker pp.1572-9575, 2019, https://doi.org/10.1007/s10773-018-3968-z
  4. A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type vol.2018, pp.2314-8888, 2018, https://doi.org/10.1155/2018/7052490
  5. Verifiable Quantum Secret Sharing Protocols Based on Four-Qubit Entangled States pp.1572-9575, 2019, https://doi.org/10.1007/s10773-019-04012-y
  6. Two-party quantum key agreement over a collective noisy channel vol.18, pp.3, 2019, https://doi.org/10.1007/s11128-019-2187-8
  7. New quantum key agreement protocols based on cluster states vol.18, pp.3, 2019, https://doi.org/10.1007/s11128-019-2200-2