We study double Hilbert transforms and maximal functions along surfaces of the form . The boundedness of the maximal operator is obtained if each is a convex increasing and . The double Hilbert transform is bounded in if both 's above are extended as even functions. If is odd, then we need an additional comparability condition on . This result is extended to higher dimensions and the general hyper-surfaces of the form on .
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