Almost orthogonal operators on the bitorus II

We prove that the operator Tf(x, y) = integral(pi)(-pi) integral(vertical bar x'vertical bar<vertical bar y'vertical bar) e(iN(x, y)x')/x' e(iN(x, y)y')/(y)' f(x - x', y - y')dx'dy', with x, y is an element of [0, 2 pi] and where the cut off vertical bar x'vertical bar < vertical bar y'vertical bar is performed in a smooth and dyadic way, is bounded from L-2 to weak-L2-epsilon, any epsilon > 0, under the basic assumption that the real-valued measurable function N(x, y) is "mainly" a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman's proof of a.e. convergence of Fourier series of L2 functions.