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

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.