Singular integrals with highly oscillating kernels on product spaces

Abstract

We prove the L 2 (T 2) boundedness of the oscillatory singular integrals P 0 f (x, y) = Dx e i(M2(x)y ′ +M1(x)x ′) x ′ y ′ f (x − x ′ , y − y ′) dx ′ dy ′ for arbitrary real-valued L ∞ functions M 1 (x), M 2 (x) and for rather general domains Dx ⊆ T 2 whose dependence upon x satisfies no regularity assumptions.