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}{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.