$L^p$ Boundedness of Hilbert Transforms Associated with Variable Plane Curves

Abstract

Let $p\in (1,\infty )$. In this paper, for any given measurable function $u:\mathbb{R}\to \mathbb{R}$ and a generalized plane curve $\gamma$ satisfying some conditions, the $L^p({\mathbb{R}}^2)$ boundedness of the Hilbert transform along the variable plane curve $u(x_1)\gamma$ $$H_{u,\gamma }f(x_1,x_2):=p.v.{\int }_{-\infty }^{\infty }f(x_1-t,x_2-u(x_1)\gamma (t))\frac{\text{d}t}{t},\forall (x_1,x_2)\in {\mathbb{R}}^2,$$ is obtained. At the same time, the $L^p(\mathbb{R})$ boundedness of the corresponding Carleson operator along the general curve $\gamma$ $${\mathcal{C}}_{u,\gamma }f(x):=p.v.{\int }_{-\infty }^{\infty }{e}^{iu(x)\gamma (t)}f(x-t)\frac{\text{d}t}{t},\forall x\in \mathbb{R},$$ is also obtained. Moreover, all the bounds are independent of the measurable function $u$.