Hilbert transforms along variable planar curves: Lipschitz regularity

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.