Singular Integrals with Bilinear Phases

We prove the boundedness from L p (T 2) to itself, 1 < p < ., 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 |y'| > |x'|, and presenting phases .(Ax+By) with 0 . A, B . 1 and . . 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A, B and . 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.