Weak type (p,q) ‐inequalities for the Haar system and differentially subordinated martingales

For any \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1\leq p,\,q<\infty$\end{document} , we determine the optimal constant \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$C_{p,q}$\end{document} such that the following holds. If \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(h_k)_{k\geq 0}$\end{document} is the Haar system on [0,1], then for any vectors a k from a separable Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal{H}$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varepsilon_k\in \{-1,1\}$\end{document} , \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$k=0,\,1,\,2,\ldots,$\end{document} we have\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \Bigg\|\sum_{k=0}^n \varepsilon_ka_kh_k\Bigg\|_{q,\infty}\leq C_{p,q}\Bigg\|\sum_{k=0}^n a_kh_k\Bigg\|_p. $$ \end{document} This is generalized to the sharp weak‐type inequality\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \|Y\|_{q,\infty}\leq C_{p,q}\|X\|_p, $$ \end{document} where X , Y stand for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal{H}$\end{document} ‐valued martingales such that Y is differentially subordinate to X .