Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Let $(h_k)_{k\geq 0}$ be the Haar system on $[0,1]$. We show that for any vectors $a_k$ from a separable Hilbert space $\mathcal{H}$ and any $\varepsilon_k\in [-1,1]$, $k=0,1,2,\ldots,$ we have the sharp inequality $$ \Bigl\|\sum_{k=0}^n \varepsilon_ka_kh_k\Big\|_{W([0,1])}\leq 2\Bigl\|\sum_{k=0}^n a_kh_k\Big\|_{L^\infty([0,1])},\quad\ n=0,1,2,\ldots,$$ where $W([0,1])$ is the weak-$L^\infty$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound $$ \|Y\|_{W(\varOmega)}\leq 2\|X\|_{L^\infty(\varOmega)},$$ where $X$ and $Y$ stand for $\mathcal{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. An application to harmonic functions on Euclidean domains is presented.