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)(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)(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),...,t(n) Gamma(t(1),...,t(n))) on R(n+1).