# Revista Matemática Iberoamericana

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**Volume 24, Issue 3, 2008, pp. 1047–1073**

**DOI: 10.4171/RMI/566**

Published online: 2008-12-31

Multiparameter singular integrals and maximal operators along flat surfaces

Yong-Kum Cho^{[1]}, Sunggeum Hong

^{[2]}, Joonil Kim

^{[3]}and Chan Woo Yang

^{[4]}(1) Chung-Ang University, Seoul, South Korea

(2) Chosun University, Gwangju, South Korea

(3) Yonsei University, Seoul, South Korea

(4) Korea University, Seoul, South Korea

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

*Keywords: *Singular Radon transform, multiple Hilbert transform, flat surface

Cho Yong-Kum, Hong Sunggeum, Kim Joonil, Yang Chan Woo: Multiparameter singular integrals and maximal operators along flat surfaces. *Rev. Mat. Iberoam.* 24 (2008), 1047-1073. doi: 10.4171/RMI/566