Bulletin of the Korean Mathematical Society (대한수학회보)

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Lp BOUNDS FOR THE PARABOLIC LITTLEWOOD-PALEY OPERATOR ASSOCIATED TO SURFACES OF REVOLUTION

  • Wang, Feixing (School of Mathematics and Physics University of Science and Technology Beijing) ;
  • Chen, Yanping (School of Mathematics and Physics University of Science and Technology Beijing) ;
  • Yu, Wei (Department of Mathematics and Mechanics Applied Science School University of Science and Technology Beijing)
  • Received : 2011.04.13
  • Published : 2012.07.31
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Abstract

In this paper the authors study the $L^p$ boundedness for parabolic Littlewood-Paley operator $${\mu}{\Phi},{\Omega}(f)(x)=\({\int}_{0}^{\infty}{\mid}F_{\Phi,t}(x){\mid}^2\frac{dt}{t^3}\)^{1/2}$$, where $$F_{\Phi,t}(x)={\int}_{p(y){\leq}t}\frac{\Omega(y)}{\rho(y)^{{\alpha}-1}}f(x-{\Phi}(y))dy$$ and ${\Omega}$ satisfies a condition introduced by Grafakos and Stefanov in [6]. The result in the paper extends some known results.

Keywords

Acknowledgement

Supported by : NSF of China

References

  1. A. Benedek, A. Calderon, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Natl. Acad. Sci. U.S.A. 48 (1962), 356-365. https://doi.org/10.1073/pnas.48.3.356
  2. 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
  3. Y. Ding, D. Fan, and Y. Pan, Lp 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
  4. Y. Ding and Y. Pan, Lp bounds for Marcinkiewicz integrals, Proc. Edinb. Math. Soc. (2) 46 (2003), no. 3, 669-677. https://doi.org/10.1017/S0013091501000682
  5. 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
  6. L. Grafakos and A. Stefanov, Convolution Calderon-Zygmund singular integral operators with rough kernel, Indiana Univ. Math. J. 47 (1998), 455-469.
  7. 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
  8. E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. https://doi.org/10.1090/S0002-9947-1958-0112932-2
  9. E. M. 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
  10. Q. Xue, Y. Ding, and K. Yabuta, Parabolic Littlewood-Paley g-function with rough kernels, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 12, 2049-2060. https://doi.org/10.1007/s10114-008-6338-6

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