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

We study the weak type inequalities for the operator $I-\mathcal {F}_s\mathcal {F}_c$, where $\mathcal {F}_c$ and $\mathcal {F}_s$ are the cosine and sine Fourier transforms on the positive half line, respectively, and $I$ is the identity operator. We also derive sharp constants in related weak type estimates for $I-\mathcal {H}^{\mathbb{T}}$, $I-\mathcal {H}^{\mathbb{R}}$ and $I-\mathcal {H}^{\mathbb{R}_+}$, where $\mathcal {H}^\mathbb{T}$, $\mathcal {H}^{\mathbb{R}}$ and $\mathcal {H}^{\mathbb{R}_+}$ denote the Hilbert transforms on the circle, on the real line and the positive half-line, respectively. Our main tool is the weak type inequality for orthogonal martingales, which is of independent interest.