Weighted weak-type inequality for martingales

Let X = (Xt)t≥0 be a bounded martingale and let Y = (Yt)t≥0 be differentially subordinate to X. We prove that if 1 ≤ p < ∞ and W = (Wt)t≥0 is an Ap weight of characteristic [W]Ap , then ||Y ||Lp,∞(W) ≤ Cp[W]Ap||X||L∞(W). The linear dependence on [W]Ap 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.