Weighted Davis Inequalities for Martingale Square Functions

Abstract

AbstractFor a Hilbert space-valued martingale $$(f_{n})$$ ( f n ) and an adapted sequence of positive random variables $$(w_{n})$$ ( w n ) , we show the weighted Davis-type inequality $$\begin{aligned} {\mathbb {E}}\left( {|}f_{0}{|} w_{0} + \frac{1}{4} \sum _{n=1}^{N} \frac{{|}df_{n}{|}^{2}}{f^{*}_{n}} w_{n} \right) \le {\mathbb {E}}( f^{*}_{N} w^{*}_{N}). \end{aligned}$$ E | f 0 | w 0 + 1 4 ∑ n = 1 N | d f n | 2 f n ∗ w n ≤ E ( f N ∗ w N ∗ ) . More generally, for a martingale $$(f_{n})$$ ( f n ) with values in a $$(q,\delta )$$ ( q , δ ) -uniformly convex Banach space, we show that $$\begin{aligned} {\mathbb {E}}\left( {|}f_{0}{|} w_{0} + \delta \sum _{n=1}^{\infty } \frac{{|}df_{n}{|}^{q}}{(f^{*}_{n})^{q-1}} w_{n} \right) \le C_{q} {\mathbb {E}}( f^{*} w^{*}). \end{aligned}$$ E | f 0 | w 0 + δ ∑ n = 1 ∞ | d f n | q ( f n ∗ ) q - 1 w n ≤ C q E ( f ∗ w ∗ ) . These inequalities unify several results about the martingale square function.