A note on the boundedness of oscillatory singular integrals with rough Kernel

This note provides a short proof that the oscillatory singular integral operator TΩ,Pf(x)=p.v.⁡∫RneiP(y)Ω(y)|y|nf(x−y) dyis bounded on L2(Rn), provided that P is a polynomial and Ω∈H1(Sn−1) satisfies the L1-Dini condition. Standard approaches to this problem typically reduce the estimates to the one-dimensional case via the van der Corput lemma. We present a different method that avoids this reduction, utilizing a direct decomposition of the frequency space in Rn with characteristic functions.