JournalsrmiVol. 24, No. 3pp. 1047–1073

Multiparameter singular integrals and maximal operators along flat surfaces

  • Yong-Kum Cho

    Chung-Ang University, Seoul, South Korea
  • Sunggeum Hong

    Chosun University, Gwangju, South Korea
  • Joonil Kim

    Yonsei University, Seoul, South Korea
  • Chan Woo Yang

    Korea University, Seoul, South Korea
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Abstract

We study double Hilbert transforms and maximal functions along surfaces of the form (t1,t2,γ1(t1)γ2(t2))(t_1,t_2,\gamma_1(t_1)\gamma_2(t_2)). The Lp(R3)L^p(\mathbb{R}^3) boundedness of the maximal operator is obtained if each γi\gamma_i is a convex increasing and γi(0)=0\gamma_i(0)=0. The double Hilbert transform is bounded in Lp(R3)L^p(\mathbb{R}^3) if both γi\gamma_i's above are extended as even functions. If γ1\gamma_1 is odd, then we need an additional comparability condition on γ2\gamma_2. This result is extended to higher dimensions and the general hyper-surfaces of the form (t1,,tn,Γ(t1,,tn))(t_1,\dots,t_{n},\Gamma(t_1,\dots,t_{n})) on Rn+1\mathbb{R}^{n+1}.

Cite this article

Yong-Kum Cho, Sunggeum Hong, Joonil Kim, Chan Woo Yang, Multiparameter singular integrals and maximal operators along flat surfaces. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 1047–1073

DOI 10.4171/RMI/566