Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A New Proof About Certain Oscillatory Singular Integrals with Nonstandard Kernel

Shen Jiawei

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 $Q(t)=\sum_{1\le i\le m} a_it^{α_i}(a_i\in\mathbb{R} \text{and } a_i\neq 0, α_i\in \mathbb{N})$ , and $Ω$ is a homogeneous function of degree zero on $\mathbb{R}^{n}$ and satisfies the vanishing moment condition. Under the condition that $Ω\in L(logL)^2(\mathbb{S}^{n-1})$ 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.

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 with nonstandard kernel defined by where , and is a homogeneous function of degree zero on and satisfies the vanishing moment condition. Under the condition that and the authors show that is bounded on with a uniform boundedness, which improves and extends the previous results.

Paper Structure

This paper contains 4 sections, 6 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prove of the Theorem \ref{['main_th']}
  4. the Proof of \ref{['L2']}

Key Result

Theorem 1

Let $\Omega$ be homogeneous of degree zero and satisfy the vanishing condition (1.1), and $A$ be a function on $\mathbb{R}^n$ such that $\nabla A\in \text{BMO}(\mathbb{R}^n)$. Suppose that $\Omega\in L(\log L)^2(S^{n - 1})$. Then, $T_{\Omega,A}$ is bounded on $L^p(\mathbb{R}^n)$ for all $p\in(1,\inf

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • proof