An Approach To Endpoint Problems in Oscillatory Singular Integrals

TL;DR

The article addresses endpoint control for maximal truncations of oscillatory singular integrals in one dimension with a polynomial phase $P$ and the Hilbert kernel. It presents an elementary approach based on pigeonholing and stationary phase to prove a weak-type $(1,1)$ bound: there exists $C_d$ such that $\| \sup_r |\int_{|t|>r} e(P(t)) f(x-t) \, dt/t| \|_{L^{1,\infty}} \le C_d \|f\|_{L^1}$ for polynomials $P$ of degree $\le d$. The proof reduces to a scale-by-scale proposition via a dyadic decomposition of $1/t$, a sparsification step excluding a small exceptional set $E$, and a modular scale selection $i \equiv m \pmod{10d}$. Core ideas include density-based interval decomposition, a light/heavy decomposition of local pieces, and a Rademacher–Menshov argument to control maximal sums across scales. Altogether, the work provides a robust, elementary endpoint estimate for a broad class of oscillatory singular integrals in 1D, laying groundwork for further endpoint analyses.

Abstract

In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.

Theorems & Definitions (3)

• Theorem 1.2
• Proposition 1.3
• Lemma 2.9: Theorem 10.6 of DTT