On a planar Pierce--Yung operator

TL;DR

This paper proves the boundedness of a specific planar operator involving a supremum and oscillatory integral on L^p spaces, answering a question posed by Pierce and Yung.

Contribution

It establishes the boundedness of a new class of oscillatory operators on L^p spaces, extending understanding of such operators in harmonic analysis.

Findings

Operator is bounded on L^p for 1<p<∞

Answers Pierce and Yung's open question

Advances the theory of oscillatory integral operators

Abstract

We show that the operator \begin{equation*} \mathcal{C} f(x,y) := \sup_{v\in \mathbb{R}} \Big|\mathrm{p.v.} \int_{\mathbb{R}} f(x-t, y-t^2) e^{i v t^3} \frac{\mathrm{d} t}{t} \Big| \end{equation*} is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p < \infty$. This gives an affirmative answer to a question of Pierce and Yung.