In the paper, we study a kind of Oscillatory singular integral operator $T_{ Q,A}$ with nonstandard Kernel defined by \[T_{ Q,A} f(x)=\text { p.v. } \int_{ \mathbb{ R}^{ n}} f(y) \frac{ \Omega(x-y)}{ |x-y|^{ n}}(A(x)-A(y)-\nabla A(x-y) e^{ i Q(|x-y|)} d y, \] where $Q(t)$ is a real polynomial on $\mathbb{ R}$, and $\Omega$ is s a homogenous function of degree zero on $\mathbb{ R}$ and satisfy satisfies the vanishing moment condition. Under the condition that $\Omega\in Llog^2(\mathbb{ R}^{ n}),$ 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.
Categories: math.CA
