A Remark on a Kind of Oscillatory Singular Integrals with Nonstarnd Kernel and Fewnomials Phases

In the paper, we study a kind of Oscillatory singular integral operator $T_{Q,A}$ with nonstandard Kernel defined by \[T_{Q,A} f(x)=\text { p.v. } \int_{\mathbb{R}^{n}} f(y) \frac{Ω(x-y)}{|x-y|^{n}}(A(x)-A(y)-\nabla A(x-y) e^{i Q(|x-y|)} d y, \] where $Q(t)$ is a real polynomial on $\mathbb{R}$, and $Ω$ is s a homogenous function of degree zero on $\mathbb{R}$ and satisfy satisfies the vanishing moment condition. Under the condition that $Ω\in Llog^2(\mathbb{R}^{n}),$ we show that $T_{Q,A}$ is bounded on $L^P(\mathbb{R}^{n})$ with uniform boundedness that depend on the number of monomials in polynomial $Q.$ which improve and extend the previous result.