Singular integrals with bilinear phases

We prove the boundedness from L-p(T-2) to itself, 1 < p < infinity, of highly oscillatory singular integrals Sf(x, y) presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by vertical bar y'vertical bar > vertical bar x'vertical bar, and presenting phases lambda(Ax+By) with 0 <= A, B <= 1 and lambda >= 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A, B and lambda involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.