\(L_ p\) inequalities for the product of the suprema of several martingales stopped at random times. Weighted norm inequalities (Q1322919)

Let \((M_ t; t \geq 0)\) and \((N_ t; t \geq 0)\) be two continuous martingales started at 0. The author considers different extensions of Burkholder-Davis (BD) inequalities and Barlow, Jacka and Yor results concerning martingales stopped at any random time. In the first part, the author compares \(E[(\sup_{t \leq K} | M_ t |^ p) (\sup_{t \leq L} | N_ t |^ p)]\) and \(E[\langle M,M \rangle_ K^{p/2} \langle N,N \rangle_ L^{p/2}]\), where \(K\) and \(L\) are any positive random variables. These two quantities are ``almost'' equivalent, we need to add a logarithmic term. Moreover these new inequalities are optimal in some sense. In the second part, the author shows BD inequalities with weight: \[ E \biggl[ \biggl( \sup_{t \leq K} | M_ t |^ p \biggr) A \biggr] \leq C \Bigl\{ E \biggl[ \bigl( \langle M,M \rangle_ K^{p \eta/2} \bigr) A \biggr] \Bigr\}^{1/ \eta}, \] \[ E \biggl[ \bigl( \langle M,M \rangle_ K^{p/2} \bigr) A \biggr] \leq C \Bigl\{ E \biggl[ \biggl( \sup_{t \leq K} | M_ t |^{p \eta} \biggr) A \biggr] \Bigr\}^{1/ \eta}, \] where \(A \geq 0\), \(E[A^{1 + \delta}] < \infty\) for some \(\delta>0\), \(\eta > 1\), \(K\) is a positive r.v., \(C\) depends on \(f,A\) and \(\eta\).