Weak-type \((1,1)\) bounds for oscillatory singular integrals with rational phases (Q2895969)

The authors consider singular integral operators \(T\) on \(\mathbb R\), defined by NEWLINE\[NEWLINETf(x)=\text{p.v.}\int_{-\infty}^\infty \mathrm e^{\mathrm iR(y)}y^{-1}f(x-y)\,dy,NEWLINE\]NEWLINE where \(R(x)=P(x)/Q(x)\) is a rational function with real coefficients. They give that \(T\) is weak-type \((1,1)\) with bounds depending only on the degrees of \(P\) and \(Q\). They comment also that it is not always the case that these operators map the Hardy space \(H^1(\mathbb R)\) to \(L^1(\mathbb R)\) and characterize those rational phases \(R(x)\) which map \(H^1(\mathbb R)\) to \(L^1(\mathbb R)\).