Sharp weighted weak type (∞, ∞) inequality for differentially subordinate martingales

Let X = (X-t)(t >= 0) be a bounded, continuous-path martingale and Y = (Y-t)(t >= 0) be a martingale that is differentially subordinate to X. We prove that if W is an A(infinity) weight of characteristic [W](A infinity), then such that parallel to Y parallel to(weak(W)) <= 97[W](A infinity) parallel to X parallel to(infinity). Here weak(W) is the weak-L-infinity space introduced by Bennett, DeVore and Sharpley. The linear dependence on [W](A infinity) is shown to be best possible. The proof exploits certain special functions enjoying appropriate size conditions and concavity. (C) 2019 Elsevier B.V. All rights reserved.