# 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

## Abstract

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 $\mathbb{R}^{n+1}$.

## 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