Sharp weighted weak-norm estimates for maximal functions

Abstract

For any 1<p<∞ and any c≥1 we identify the least constant Cp,c with the following property. If X=(Xt)t≥0 is a uniformly integrable martingale and W=(Wt)t≥0 is a weight satisfying Muckenhoupt’s condition Ap with [W]Ap≤c, then we have the Lorentz-norm estimate ||supt≥0|Xt|||Lp,∞(W)≤Cp,c‖X∞‖Lp,∞(W).The proof exploits related sharp weak-type estimates and optimization arguments.