$L^2$ Boundedness of Hilbert Transforms along Variable Flat Curves

TL;DR

This paper establishes the $L^2$ boundedness of Hilbert transforms along variable flat curves in the plane, introducing a new condition on the curve $oldsymbol{ extgamma}$ that ensures boundedness.

Contribution

It provides a novel sufficient condition on the curve $oldsymbol{ extgamma}$ for the $L^2$ boundedness of Hilbert transforms along variable flat curves.

Findings

Proves $L^2$ boundedness under new curve condition

Introduces a sufficient condition on $ extgamma$

Extends understanding of variable flat curve transforms

Abstract

In this paper, the $L^2$ boundedness of the Hilbert transform along variable flat curve $(t,P(x_1)\gamma(t))$ $$H_{P,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-P(x_1)\gamma(t))\,\frac{\textrm{d}t}{t},\quad \forall\, (x_1,x_2)\in\mathbb{R}^2,$$ is studied, where $P$ is a real polynomial on $\mathbb{R}$. A new sufficient condition on the curve $\gamma$ is introduced.