On the best constants in the weak type inequalities for re-expansion operator and Hilbert transform (Q2841355)

The cosine and sine Fourier transforms, \(\mathcal{F}_c\) and \(\mathcal{F}_s\), and the Hilbert transforms on the circle, on the real line and on the positive half-line, \(\mathcal{H}^{\mathbb{T}}\), \(\mathcal{H}^{\mathbb{R}}\) and \(\mathcal{H}^{\mathbb{R}_+}\), are investigated. Sharp constants of the weak type \(L_p\to L_{p,\infty}\) inequality are derived for the operators \(I-\mathcal{F}_s\mathcal{F}_c\), \(I-\mathcal{H}^{\mathbb{T}}\), \(I-\mathcal{H}^{\mathbb{R}}\) and \(I-\mathcal{H}^{\mathbb{R}_+}\), respectively, where \(I\) is the identity operator and \(L_{p,\infty}\) the weak \(L_p\) space. The main tool of the proof is the weak type inequality for orthogonal martingales.