Weighted Davis Inequalities for Martingale Square Functions

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