Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let (M,τ) be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras (M_t)_0≤t≤1 whose union is weak-⁎ dense in M. Let E_t denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for 1<p<2 and x∈L_p(M,τ) one can find a,b∈L_p(M,τ) and contractions u_t,v_t∈M such that E_t(x)=au_t+v_tbandmax⁡{‖a‖_p,‖b‖_p}≤c_p‖x‖_p. Moreover, au_t and v_tb converge in the row/column Hardy spaces H_p^r(M) and H_p^c(M) respectively. We also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations, including p=1. As for other results in noncommutative martingale theory, the passage from discrete to continuous index is quite technical and requires genuinely new methods. Our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of L_p-modules. The convergence in H_p^r(M) and H_p^c(M) also imposes new algebraic atomic decompositions.