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

Abstract:

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 \[ \left |\left |\sum _{k=0}^n \theta _ka_kh_k\right |\right |_{p,\infty }\leq C_p\left |\left |\sum _{k=0}^n a_kh_k\right |\right |_p.\] This is generalized to the weak type inequality \[ ||g||_{p,\infty }\leq C_p||f||_p,\] 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 \[ ||Y||_{p,\infty }\leq C_p||X||_p,\] 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 $t$.