Weighted estimates for oscillatory singular integrals (Q1951068)

Summary: We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy space \(H^1(\mathbb S^{n-1})\), while such results were known previously only for kernels in \(L \log L(\mathbb S^{n-1})\), a proper subspace of \(H^1(\mathbb S^{n-1})\). One of our results establishes a \(L^p(w) \to L^p(w)\) bound for certain weights. At the same time, it provides a solution to an open problem stated in [\textit{S. Lu}, Int. J. Appl. Math. Sci. 2, No. 1, 47--64 (2005; Zbl 1078.42006)].