Weighted estimates for oscillatory singular integrals

Abstract

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 H1(Sn-1), while such results were known previously only for kernels in L log L(Sn-1), a proper subspace of H1(Sn-1). One of our results established a Lp(w)→Lp(w) bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005). © 2013 H. Al-Qassem et al.