# The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals

## TL;DR

This paper advances understanding of the boundedness of maximal modulations of the Hilbert transform along the parabola by combining polynomial Carleson theorem techniques with oscillatory decay methods.

## Contribution

It establishes uniform $L^p$ bounds for maximal modulations of the Hilbert transform along the parabola, integrating polynomial Carleson theorem and oscillatory decay strategies.

## Key findings

- Proved uniform $L^p$ bounds for maximal modulations
- Identified effective use of polynomial Carleson theorem
- Utilized oscillation for decay via $TT^*$ method

## Abstract

We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the $TT^*$ method.