Remarks on the maximal operator and Hilbert transform along variable parabolas

TL;DR

This paper establishes boundedness results for the maximal operator and Hilbert transform along variable parabolas in L^p spaces for p>p_0, exploring their connections with vector field Hilbert transforms and polynomial Carleson operators.

Contribution

It proves boundedness of these operators along variable parabolas in L^p spaces and discusses their relation to other harmonic analysis operators.

Findings

Boundedness of maximal operator along variable parabolas for p>p_0

Boundedness of Hilbert transform along variable parabolas for p>p_0

Connections with Hilbert transform along vector fields and polynomial Carleson operator

Abstract

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's maximal operator are also discussed.