Weak Type Bound for Oscillatory Singular Integrals

TL;DR

This paper establishes a weak type (1,1) bound for oscillatory singular integrals with polynomial phase, simplifying and extending previous proofs for maximal truncations of such operators.

Contribution

It provides a simplified and extended proof of the weak L^1 inequality for maximal truncations of oscillatory singular integrals with polynomial phase.

Findings

Proves weak L^1 bound for oscillatory singular integrals

Extends previous results to broader class of operators

Simplifies the proof technique used in earlier work

Abstract

Let $ T _{P} f (x) = \int e ^{i P (y)} K (y) f (x-y) \, dy $, where $ K (y)$ is a smooth Calder\'on-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. The maximal truncations of $ T_P$ satisfy the weak $ L ^{1}$ inequality, our proof simplifying and extending the argument of Chanillo and Christ for the weak type bound for $ T_P$.