Oscillatory Singular Integrals with Rough Kernel

Abstract

This paper is devoted to the study on the L p -boundedness for the oscillatory singular integral defined by $$ Tf(x) = p.v.\int_{\mathbb{R}^n } {e^{iP(x,y)} } K(x - y)f(y)dy, $$ where P(x,y) is a real polynomial on ℝ n × ℝ n , and \( K(x) = \frac{{h(\mid x\mid \Omega (x)}} {{\mid x\mid ^n }} \) with Ω ∈ Llog + L(S n−1) and h ∈ BV(ℝ+) (i.e. h is a bounded variation function on ℝ+).