Weak-type estimates for martingale maximal functions

Abstract

The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair (x, y) of nonnegative numbers satisfying x p ≤ y, we identify the optimal upper bounds for | sup n fn| p,∞, for nonnegative martingales f = (fn) n≥0 satisfying f 1 = x and f p p = y.