Multiparameter singular integrals and maximal operators along flat surfaces

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