$L^p$ boundness of Oscillatory singular integral with Calder\'{o}n Type Commutators

Abstract

In the paper, we study a kind of Oscillatory singular integral operator with Calder\'{o}n Type Commutators $T_{P,K,A} $ defined by \[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y) e^{i P(x-y)} d y, \] where $P(t)$ is a real polynomial on $\mathbb{R},$ and $K$ is a function on $\mathbb{R}^{n},$ satisfies the vanishing moment and $CZ(\delta)$ conditions. Under these conditions, we show that $T_{P,K,A}$ is bounded on $L^p(\mathbb{R}^{n})$ with uniform boundedness, which improve and extend the previous result.