Uniform bounds for oscillatory and polynomial Carleson operators

TL;DR

This paper establishes uniform bounds for a class of oscillatory and polynomial Carleson operators, including near a quadratic Carleson operator, advancing understanding of their boundedness properties.

Contribution

It introduces techniques to prove uniform bounds for oscillatory and polynomial Carleson operators, including near a single scale quadratic case, which was previously unresolved.

Findings

Proved uniform bounds for oscillatory Carleson operators.

Extended bounds to polynomial Carleson operators.

Applied techniques to near a single scale quadratic Carleson operator.

Abstract

We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson operators near a single scale version of the quadratic Carleson operator.