  We establish distributional estimates for noncommutative martingales, in the
sense of decreasing rearrangements of the spectra of unbounded operators, which
generalises the study of distributions of random variables. Our results include
distributional versions of the noncommutative Stein, dual Doob, martingale
transform and Burkholder-Gundy inequalities. Our proof relies upon new and
powerful extrapolation theorems. As an application, we obtain some new
martingale inequalities in symmetric quasi-Banach operator spaces and some
interesting endpoint estimates. Our main approach demonstrates a method to
build the noncommutative and classical probabilistic inequalities in an
entirely operator theoretic way.
