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 $1<p<\mathrm{\infty }$ and a pair $(x,y)$ of nonnegative numbers satisfying $x^p\le y$, we identify the optimal upper bounds for $\Vert |{sup}_n{f}_n|{\Vert }_{p,\mathrm{\infty }}$, for nonnegative martingales $f={(f_n)}_{n\ge 0}$ satisfying $\Vert f{\Vert }_1=x$ and $\Vert f{\Vert }_p^p=y$.