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

Abstract

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