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 \[T_{Q,A} f(x)=\text { p.v. } \int_{\mathbb{R}^{n}} f(y) \frac{\Omega(x-y)}{|x-y|^{n+1}}\left(A(x)-A(y)-\nabla A(y)(x-y)\right) e^{i Q(|x-y|)} d y, \] where $Q(t)=\sum_{1\le i\le m} a_it^{\alpha_i}(a_i\in\mathbb{R} \text{and } a_i\neq 0, \alpha_i\in \mathbb{N})$ , and $\Omega$ is a homogeneous function of degree zero on $\mathbb{R}^{n}$ and satisfies the vanishing moment condition. Under the condition that $\Omega\in L(logL)^2(\mathbb{S}^{n-1})$ 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.