Maximal Polynomial Modulations of Singular Radon Transforms

TL;DR

This paper establishes $L^2 o L^p$ bounds for maximal polynomial modulations of Calderón-Zygmund operators on the torus, improving constants and showing boundedness with logarithmic dependence for mollified Hilbert transforms along the parabola.

Contribution

It provides new $L^2 o L^p$ estimates for polynomial modulated singular integrals with anisotropic scaling, including improved constants and applications to mollified Hilbert transforms.

Findings

Established $L^2 o L^p$ bounds on the torus for maximal polynomial modulations.

Achieved improved constants in the estimates.

Proved boundedness of mollified Hilbert transforms along the parabola with logarithmic dependence.

Abstract

We prove $L^2 \to L^p$ estimates on the torus for maximal polynomial modulations of Calder\'on-Zygmund operators with anisotropic scaling. We obtain improved constants in these estimates. As a corollary, maximal polynomial modulations of a mollified version of the Hilbert transform along the parabola are bounded with only logarithmic dependence of the estimate on the Lipschitz constant of the mollifier.