Remarks on the class of continuous martingales with bounded quadratic variation (Q1059926)

For every continuous local martingale M let \(<M>\) as usual denote the quadratic variation of M. The space \(H^{\infty}\) of all M such that \(<M>_{\infty}\) is a.s. bounded is clearly contained in the space BMO. Another subspace of BMO is the space \(L^{\infty}\) of all bounded martingales. The present paper gives some new informations about the relations between \(L^{\infty}\) and \(H^{\infty}\) and especially about \(H^{\infty}\) as a subspace of BMO. It is proved that \(H^{\infty}\) has a property connected with weighted norm inequalities for martingales. As a consequence one gets that BMO\(\setminus \overline{H^{\infty}}\neq \emptyset\), if \(BMO\neq L^{\infty}\), and that \(L^{\infty}\) is not contained in \(\overline{H^{\infty}}\).