A New Proof About Certain Oscillatory Singular Integrals with Nonstandard Kernel

In the paper, we provide a new method to study the oscillatory singular integral operator $T_{Q,A}$ with nonstandard kernel defined by $$T_{Q,A}f(x)=\text{ p.v. }{\int }_{{\mathbb{R}}^n}f(y)\frac{\mathrm{\Omega }(x-y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+1}}(A(x)-A(y)-\mathrm{\nabla }A(y)(x-y))e^{iQ(\mathrm{\mid }x-y\mathrm{\mid })}dy,$$ where $Q(t)={\sum }_{1\le i\le m}{a}_i{t}^{{\alpha }_i}(a_i\in \mathbb{R}\text{and }{a}_i\mathrm{\ne }0,{\alpha }_i\in \mathbb{N})$ , and $\mathrm{\Omega }$ is a homogeneous function of degree zero on ${\mathbb{R}}^n$ and satisfies the vanishing moment condition. Under the condition that $\mathrm{\Omega }\in L(logL{)}^2({\mathbb{S}}^{n-1})$ and $\mathrm{\nabla }A\in \text{BMO}({\mathbb{R}}^n),$ the authors show that $T_{Q,A}$ is bounded on $L^p({\mathbb{R}}^n)$ with a uniform boundedness, which improves and extends the previous results.