Almost orthogonal operators on the bitorus II

We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L ² to weak- ${L^{2-\epsilon}}$ , 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 L ² functions.