Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

Lp BOUNDS FOR THE PARABOLIC MARCINKIEWICZ INTEGRAL WITH ROUGH KERNELS

  • Chen, Yanping (SCHOOL OF MATHEMATICAL SCIENCES BEIJING NORMAL UNIVERSITY, APPLIED SCIENCE SCHOOL UNIVERSITY OF SCIENCE AND TECHNOLOGY BEIJING) ;
  • Ding, Yong (SCHOOL OF MATHEMATICAL SCIENCES BEIJING NORMAL UNIVERSITY)
  • Published : 2007.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We prove the $L^p(1 boundedness of the parabolic Marcinkiewicz integral with the kernel function $\Omega{\in}L(log^+L)^{1/2}(S^{n-1})$. The result is an improvement and extension of some known results.

Keywords

References

  1. A. AL-Salman, H. AL-Qassem, L. Cheng, and Y. Pan, LP 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
  2. A. Benedek, A. Calderon, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356-365 https://doi.org/10.1073/pnas.48.3.356
  3. A. Calderon and A. Zygmund, A note on the interpolation of sublinear operations, Amer. J. Math. 78 (1956), 282-288 https://doi.org/10.2307/2372516
  4. Y. Ding, D. Fan, and Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J. 48 (1999), no. 3, 1037-1055
  5. Y. Ding, D. Fan, and Y. Pan, $L^p$-boundedness of Marcinkiewicz integrals with Hardy space function kernels, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 4, 593-600 https://doi.org/10.1007/s101140000015
  6. Y. Ding, Q. Xue and K. Yabuta, Parabolic Littlewood-Paley g-function with rough kernels, preprint
  7. J. Duoandikoetxea and J. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541-561 https://doi.org/10.1007/BF01388746
  8. E. Fabes and N. Riviere, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19-38 https://doi.org/10.4064/sm-27-1-19-38
  9. W. R. Madych, On Littlewood-Paley functions, Studia Math. 50 (1974), 43-63 https://doi.org/10.4064/sm-50-1-43-63
  10. A. Nagel, N. Riviere and S. Wainger, On Hilbert transforms along curves. II., Amer. J. Math. 98 (1976), no. 2, 395-403 https://doi.org/10.2307/2373893
  11. F. Ricci and E. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 637-670 https://doi.org/10.5802/aif.1304
  12. E. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958) 430-466 https://doi.org/10.2307/1993226
  13. E. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239-1295 https://doi.org/10.1090/S0002-9904-1978-14554-6
  14. T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203-217 https://doi.org/10.4064/sm-44-3-203-217

Cited by

  1. Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces vol.2014, pp.1, 2014, https://doi.org/10.1186/1029-242X-2014-265
  2. BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES vol.99, pp.03, 2015, https://doi.org/10.1017/S1446788715000191
  3. Lp boundedness for commutators of parabolic Littlewood-Paley operators with rough kernels vol.284, pp.8-9, 2011, https://doi.org/10.1002/mana.200810139
  4. PARABOLIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO POLYNOMIALS COMPOUND CURVES AND EXTRAPOLATION vol.52, pp.3, 2015, https://doi.org/10.4134/BKMS.2015.52.3.771
  5. Parabolic Marcinkiewicz integrals along surfaces on product domains vol.27, pp.1, 2011, https://doi.org/10.1007/s10114-010-9653-7
  6. L p Bounds for the Commutator of Parabolic Singular Integral with Rough Kernels vol.27, pp.4, 2007, https://doi.org/10.1007/s11118-007-9062-4
  7. Rough Marcinkiewicz integrals with mixed homogeneity on product spaces vol.29, pp.7, 2013, https://doi.org/10.1007/s10114-013-1675-5
  8. Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces vol.27, pp.1, 2011, https://doi.org/10.1007/s10496-011-0059-x