  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.
