A Four-Weight Weak Type Maximal Inequality for Martingales

Abstract

In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form

$$\int_{\left\{ {f* > \lambda } \right\}} {\Phi _1 \left( {\lambda w_1 } \right)w_2 d\mathbb{P}} \leqslant C\int_\Omega {\Phi _2 \left( {C\left| {f_\infty } \right|w_3 } \right)w_4 d} \mathbb{P}$$

holds a.e. for uniformly integrable martingales f = (f_n)_n≥0 with some constant C > 0, where Φ₁, Φ₂ are Young functions, w_i (i = 1, 2, 3, 4) are weights, $f* = \mathop {\sup }\limits_{n \geqslant 0} \left| {f_n } \right|$ and $f_\infty = \mathop {\lim }\limits_{n \to \infty } f_n$ a.e. As an application, two-weight weak type maximal inequalities of martingales are considered, and particularly a new equivalence condition is presented.