$L^p$ boundness of Oscillatory singular integral with Calder\'{o}n Type Commutators

TL;DR

This paper proves the boundedness of a class of oscillatory singular integral operators with Calderón type commutators on L^p spaces, extending previous results by establishing uniform bounds under certain conditions.

Contribution

The paper introduces new boundedness results for oscillatory singular integrals with Calderón type commutators, generalizing prior work with broader conditions.

Findings

Boundedness of $T_{P,K,A}$ on $L^p$ spaces for $1<p< abla$

Uniform boundedness under vanishing moment and $CZ( abla)$ conditions

Extension of previous boundedness results to more general operators

Abstract

In the paper, we study a kind of Oscillatory singular integral operator with Calder\'{o}n Type Commutators $T_{P,K,A} $ defined by \[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y) e^{i P(x-y)} d y, \] where $P(t)$ is a real polynomial on $\mathbb{R},$ and $K$ is a function on $\mathbb{R}^{n},$ satisfies the vanishing moment and $CZ(\delta)$ conditions. Under these conditions, we show that $T_{P,K,A}$ is bounded on $L^p(\mathbb{R}^{n})$ with uniform boundedness, which improve and extend the previous result.