Operators on martingales, ф-summing operators, and the contraction principle

For the absolutely Ф-summmg operators T: X→Y between Banach spaces X and Y we consider martingale inequalities of the type…[formula] where ..[formula]…is a martingale difference sequence and i is a sequence of normalized functionals on X, and we show that these inequalities are useful in different directions. For example, for a Banach space X, x1…xn ∈X, independent standard Gaussian variables gn, and 1 < r < ∞ we deduce that..[formula]… where is a scale-valued martingale difference sequence such that [formula]…is predictable ..[formula].. is a sequence of stopping times and [formula].