Sharp weak type inequalities for the Haar system and related estimates for nonsymmetric martingale transforms

For any $1\leq p<\infty$, we determine the optimal constant $C_p$ such that the following holds. If $(h_k)_{k\geq 0}$ is the Haar system, then for any vectors $a_k$ from a separable Hilbert space $\mathcal {H}$ and $\theta _k\in \{0,1\}$, $k=0,\,1,\,2,\,\ldots$, we have

$$\left \vert\left \vert\sum _{k=0}^n \theta _ka_kh_k\right \vert\r... ...}\leq C_p\left \vert\left \vert\sum _{k=0}^n a_kh_k\right \vert\right \vert _p.$$

This is generalized to the weak type inequality

$$\vert\vert g\vert\vert _{p,\infty }\leq C_p\vert\vert f\vert\vert _p,$$

where $f$ is an $\mathcal {H}$-valued martingale and $g$ is its transform by a predictable sequence taking values in $[0,1]$. We extend this further to the estimate

$$\vert\vert Y\vert\vert _{p,\infty }\leq C_p\vert\vert X\vert\vert _p,$$

valid for any two $\mathcal {H}$-valued continuous-time martingales $X$, $Y$, such that $([Y,X-Y]_t)$ is nondecreasing and nonnegative as a function of $t$.