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

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}{2π}|\{ ξ\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(ξ) \geq 1 \}| \leq \frac{4}π\arctan\left(\exp\left(\fracπ{2}\|f\|_1\right)\right) -1, \quad f\in L^{1}(\mathbb{T}), \end{equation*} and \begin{equation*} \frac{1}{2π}|\{ ξ\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(ξ) \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.

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