On Fefferman and Burkholder-Davis-Gundy inequalities for E-martingales

In a previous paper we introduced a new concept, the notion of E-martingale s and we extended the well-known Doob inequality (for 1 < p < +infinity) an d the Burkholder-Davis-Gundy inequalities (for p 2) to E-martingales. After showing new Fefferman-type inequalities that involve sharp brackets as wel l as the space bmo(q), we extend the Burkholder-Davis-Gundy inequalities (f or 1 < p < +infinity) to E-martingales. By means of these inequalities we g ive sufficient conditions for the closedness in L-p of a space of stochasti c integrals with respect to a fixed R-d-valued semimartingale, a question w hich arises naturally In the applications to financial mathematics. Finally we investigate the relation between uniform convergence in probability and semimartingale topology.