A note on sharp one-sided bounds for the Hilbert transform (Q2790193)

The author establishes a sharp upper bound for the value NEWLINE\[NEWLINE \frac{1}{2\pi} \left|\left\{\xi\in {\mathbb T}: {\mathcal H}^{\mathbb T}f(\xi)\geq 1\right\}\right| = :{\mathcal M}(f), NEWLINE\]NEWLINE where \({\mathcal H}^{\mathbb T}\) stands for the Hilbert transform on the unit circle \({\mathbb T}\). In particular, he shows that NEWLINE\[NEWLINE{\mathcal M}(f)\leq \frac{4}{\pi} \arctan\left(\exp\left(\frac{\pi}{2}\|f\|_{1}\right)\right)-1 NEWLINE\]NEWLINE for \(f\in L^{1}({\mathbb T})\), and NEWLINE\[NEWLINE{\mathcal M}(f)\leq \frac{\|f\|^{2}_{2}}{1+ \|f\|^{2}_{2}} NEWLINE\]NEWLINE for \(f\in L^{2}({\mathbb T})\). Here \(\|f\|_{1,2}\) are the norms with respect to the normalized Haar measure on the circle.