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

TL;DR

This paper proves the boundedness of certain Hilbert transforms and Carleson operators along variable plane curves in $L^p$ spaces, with bounds independent of the variable function $u$, extending classical harmonic analysis results.

Contribution

It establishes $L^p$ boundedness for Hilbert transforms and Carleson operators along variable plane curves, generalizing previous fixed-curve results with bounds independent of the measurable function $u$.

Findings

Proved $L^p$ boundedness of Hilbert transforms along variable curves.

Established $L^p$ boundedness of Carleson operators with variable phase.

Bounds are independent of the measurable function $u$.

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