Weak-type (1,1) bounds for oscillatory singular integrals with rational phases

We consider singular integral operators on R given by convolution with a principal value distribution defined by integrating against oscillating kernels of the form e(iR(x))/x where R(x) = P(x)/Q(x) is a general rational function with real coefficients. We establish weak-type (1,1) bounds for such operators which are uniform in the coefficients, depending only on the degrees of P and Q. It is not always the case that these operators map the Hardy space H-1(R) to L-1(R) and we will characterise those rational phases R(x) = P(x)/Q(x) which do map H-1 to L-1 (and even H-1 to H-1).