Weighted weak-type inequality for martingales

Let $X=(X_t)_{t\geq 0}$ be a bounded martingale and let $Y=(Y_t)_{t\geq 0}$ be differentially subordinate to $X$. We prove that if $1\leq p \lt \infty $ and $W=(W_t)_{t\geq 0}$ is an $A_p$ weight of characteristic $[W]_{A_p}$, then $$ \| Y\| _{L^{p,\infty }(W)}\leq C_p[W]_{A_p}\| X\| _{L^\infty (W)}.$$ The linear dependence on $[W]_{A_p}$ is shown to be the best possible. The proof exploits a weighted exponential bound which is of independent interest. As an application, a related estimate for the Haar system is established.