Hilbert transforms along variable planar curves: Lipschitz regularity

TL;DR

This paper proves the boundedness of Hilbert transforms along variable planar curves with Lipschitz regularity, extending classical results to more general, variable geometric settings.

Contribution

It establishes $L^p$-boundedness for Hilbert transforms along variable curves with Lipschitz regularity, a novel extension of classical harmonic analysis results.

Findings

$L^p$-boundedness of $H^{mma}$ for $1<p<ty$

Boundedness holds for curves with small Lipschitz norm

Results apply to general curves with smoothness and curvature conditions

Abstract

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1, x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $\gamma$ is a general curve satisfying some suitable smoothness and curvature conditions.