$L^p$ boundedness of oscillatory singular integrals with nonstandard kernels

In this paper, we study a kind of oscillatory singular integral operator $T_{P,\mathrm{\Omega },A}$ defined by

$$T_{P,\mathrm{\Omega },A}f(x)=\text{p.v.}{\int }_{{\mathbb{R}}^n}{e}^{iP(x-y)}\frac{\mathrm{\Omega }(x-y)\left[A(x)-A(y)-\mathrm{\nabla }A(y)\cdot (x-y)\right]}{|x-y{|}^{n+1}}f(y)dy,$$

where $P(x)$ is a real polynomial on ${\mathbb{R}}^n,$ and $\mathrm{\Omega }(\lambda x)=\mathrm{\Omega }(x).$ Under some conditions, we show that $T_{P,\mathrm{\Omega },A}$ is bounded on $L^p({\mathbb{R}}^n)$ with uniform boundedness, which improves and extends Pan's result ^[11].