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 $\mathbb{R}^{n+1}$.
Citation
Yong-Kum Cho . Sunggeum Hong . Joonil Kim . Chan Woo Yang . "Multiparameter singular integrals and maximal operators along flat surfaces." Rev. Mat. Iberoamericana 24 (3) 1047 - 1073, November, 2008.
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