Illinois Journal of Mathematics

The boundedness of Marcinkiewicz integral with variable kernel

Chin-Cheng Lin, Ying-Chieh Lin, Xiangxing Tao, and Xiao Yu

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In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by $$\mu_{\Omega,\alpha}(f)(x)=\bigg(\int_{0}^{\infty}\bigg| {\int_{|x-y|\leq t}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}f(y)\,dy\bigg|^{2} \frac{dt}{t^{3-\alpha}}\bigg)^{1/2}, %\\ %\quad0<\alpha\leq2. $$where $0<\alpha\leq2$. We first prove that $\mu_{\Omega,\alpha}$ is bounded from $L^{{2n}/{n+\alpha}}(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$ without any smoothness assumption on the kernel $\Omega$. Then we show that, if the kernel $\Omega$ satisfies a class of Dini condition, $\mu_{\Omega,\alpha}$ is bounded from $H^p(\mathbb{R}^n)$ ($p \le1$) to $H^q(\mathbb{R}^n)$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{2n}$. As corollary of the above results, we obtain the $L^p-L^q$ ($1< p<2$) boundedness of this fractional Marcinkiewicz integral.

Article information

Illinois J. Math., Volume 53, Number 1 (2009), 197-217.

First available in Project Euclid: 22 January 2010

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Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces


Lin, Chin-Cheng; Lin, Ying-Chieh; Tao, Xiangxing; Yu, Xiao. The boundedness of Marcinkiewicz integral with variable kernel. Illinois J. Math. 53 (2009), no. 1, 197--217. doi:10.1215/ijm/1264170846.

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  • N. E. Aguilera and E. O. Harboure, Some inequalities for maximal operators, Indiana Univ. Math. J. 29 (1980), 559–576.
  • A. Benedek, A. P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. USA 48 (1962), 356–365.
  • M. Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc. 133 (2005), 3535–3542.
  • M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003).
  • A. P. Calderón and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc. 78 (1955), 209–224.
  • ––––, On singular integrals with variable kernels, Appl. Anal. 7 (1977/78), 221–238.
  • Y. Ding, J. Chen and D. Fan, A class of integral operators with variable kernels on Hardy space, Chinese Ann. Math. Ser. A 23 (2002), 289–296.
  • Y. Ding, M.-Y. Lee and C.-C. Lin, Marcinkiewicz integral on weighted Hardy spaces, Arch. Math. 80 (2003), 620–629.
  • Y. Ding, C.-C. Lin and Y.-C. Lin, Erratum: “On Marcinkiewicz integral with variable kernels", Vol. 53 (2004), 805–821, Indiana Univ. Math. J. 56 (2007), 991–994.
  • Y. Ding, C.-C. Lin and S. Shao, On the Marcinkiewicz integral with variable kernels, Indiana Univ. Math. J. 53 (2004), 805–821.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Publishing Co., Amsterdam, 1985.
  • L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.
  • L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93–140.
  • M.-Y. Lee and C.-C. Lin, Weighted $L^p$ boundedness of Marcinkiewicz integral, Integr. Equ. Oper. Theory 49 (2004), 211–220.
  • M.-Y. Lee, C.-C. Lin, Y.-C. Lin and D. Yan, Boundedness of singular integral operators with variable kernels, J. Math. Anal. Appl. 348 (2008), 787–796.
  • C.-C. Lin and Y.-C. Lin, $H^p_w-L^p_w$ boundedness of Marcinkiewicz integral, Integr. Equ. Oper. Theory 58 (2007), 87–98.
  • J. Marcinkiewicz, Sur quelques intégrales du type de Dini, Ann. Soc. Pol. Math. 17 (1938), 42–50.
  • S. Meda, P. Sjögren and M. Vallarino, On the $H^1-L^1$ boundedness of operators, Proc. Amer. Math. Soc. 136 (2008), 2921–2931.
  • Y. Meyer and R. Coifman, Wavelets. Calderón–Zygmund and multilinear operators, Cambridge Univ. Press, Cambridge, 1997.
  • M. Sakamoto and K. Yabuta, Boundedness of Marcinkiewicz functions, Studia Math. 135 (1999), 103–142.
  • E. M. Stein, On the function of Littlewood–Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466.
  • E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971.
  • K. Yabuta, A remark on the $(H^1, L^1)$ boundedness, Bull. Fac. Sci., Ibaraki Univ. Ser. A 25 (1993), 19–21.
  • A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170–204.