Theory of H_p-spaces for continuous filtrations in von neumann algebras

We introduce Hardy spaces for martingales with respect to continuous filtration for von Neumann algebras. In particular we prove the analogues of the Burkholder-Gundy and Burkholder-Rosenthal inequalities in this setting. The usual arguments using stopping times in the commutative case are replaced by tools from noncommutative function theory and allow us to obtain the analogue of the Feffermann-Stein duality and prove a noncommutative Davis decomposition.