A note on sharp one-sided bounds for the Hilbert transform

TL;DR

This paper establishes sharp one-sided bounds for the Hilbert transform on the circle in terms of L^1 and L^2 norms, and extends related estimates to orthogonal martingales with subordination.

Contribution

It provides the first sharp bounds for the Hilbert transform's level set measures in terms of L^1 and L^2 norms, and relates these to martingale inequalities.

Findings

Sharp bounds for Hilbert transform on the circle in L^1 and L^2 norms.

Extension of bounds to orthogonal martingales with subordination.

Explicit formulas involving arctangent and quadratic fractions.

Abstract

Let $\mathcal{H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{4}{\pi}\arctan\left(\exp\left(\frac{\pi}{2}\|f\|_1\right)\right) -1, \quad f\in L^{1}(\mathbb{T}), \end{equation*} and \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{\|f\|_2^2}{1+\|f\|_2^2},\quad f\in L^{2}(\mathbb{T}). \end{equation*} Related estimates for orthogonal martingales satisfying a subordination condition are also established.