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.