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 λ(Ax+By) with 0 ≤ A, B ≤ 1 and λ ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A, B and λ 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. Supported by Progetto cofimansiato HIUR "Amatisi Armomica"