A New Proof About Certain Oscillatory Singular Integrals with Nonstandard Kernel

Abstract

In the paper, we provide a new method to study the oscillatory singular integral operator $T_{Q,A}$ with nonstandard kernel defined by \[T_{Q,A} f(x)=\text { p.v. } \int_{\mathbb{R}^{n}} f(y) \frac{Ω(x-y)}{|x-y|^{n+1}}\left(A(x)-A(y)-\nabla A(y)(x-y)\right) e^{i Q(|x-y|)} d y, \] where $Q(t)={\sum }_{1\le i\le m}{a}_i{t}^{\hat{I}{\pm }_i}(a_i\in \mathbb{R}\text{and }{a}_i\ne 0,\hat{I}{\pm }_i\in \mathbb{N})$ , and $\hat{I}\text{c◯}$ is a homogeneous function of degree zero on ${\mathbb{R}}^n$ and satisfies the vanishing moment condition. Under the condition that $\hat{I}\text{c◯}\in L(logL{)}^2({\mathbb{S}}^{n-1})$ and $\nabla A\in \text{BMO}({\mathbb{R}}^n),$ 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.