A sharp weak-type \((\infty,\infty)\) inequality for the Hilbert transform (Q317795)

The notation \(\mathbb{S}\) stands for the circle \(\mathbb{T}\) or the real line \(\mathbb{R}\) and \(W(\mathbb{S})\) denotes the weak-\(L^\infty\) space introduced by Bennett, DeVore and Sharpley. Then the author proved the following estimate: \[ \| \mathcal{H}^{\mathbb{S}}f\|_{W(\mathbb{S})}\leq \|f\|_{L^\infty(\mathbb{S})}, \] where \(\mathcal{H}^{\mathbb{S}}\) is the Hilbert transform on \(\mathbb{S}\). The constant 1 is shown to be the best possible.