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 ℝⁿ × ℝⁿ, 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₊).

Let $\overline {T_{} }$ be a singular integral operator corresponding to T, and let $\overline {T_o }$ be the truncated operator of $\overline {T_{} }$. That means

$$ \overline {T_0 } f(x) = p.v.\int_{|x - y| < 1} {K(x - y)f(y)dy.} $$

The main result in this paper gives out a verifiable necessary and sufficient condition on $\overline {T_o }$ so that the oscillatory integral operator T is bounded on L ^P(ℝⁿ), 1 <p<∞, for any real non-trivial polynomial P(x,y. In addition, we also discuss the weighted L ^P-boundedness of T.

The Project is supported by National Natural Science Foundation of China