Noncommutative Burkholder/Rosenthal inequalities. (English) Zbl 1041.46050
This is the third in a series of papers that pushed the non-commutative martingale theory to new limits, the first two being the paper of G. Pisier and Q. Xu on Burkholder-Gundy inequalities in the tracial case [Commun. Math. Phys. 189, No. 3, 667–698 (1997; Zbl 0898.46056)] and that of M. Junge on Doob’s inequalities in the non-tracial case [J. Reine Angew. Math. 549, 149–190 (2002; Zbl 1004.46043)].
First, the authors generalize the Burkholder-Gundy inequalities to the non-tracial case. This makes it possible to give a description of the duals of the Hardy spaces \(H^p\) as the spaces of martingale difference sequences in \(L^q\) of bounded mean oscillation (recovering BMO for \(q=\infty\)). Secondly, they prove the Burkholder martingale inequalities with the conditioned square function (of which the Rosenthal inequality is a special case) in full generality. Thirdly, they extend the dual form of Doob’s inequality from Junge’s paper (by reversing it) to the values \(0<p<1\). Finally, the non-faithful case and the order of growth of the involved constants are discussed.
First, the authors generalize the Burkholder-Gundy inequalities to the non-tracial case. This makes it possible to give a description of the duals of the Hardy spaces \(H^p\) as the spaces of martingale difference sequences in \(L^q\) of bounded mean oscillation (recovering BMO for \(q=\infty\)). Secondly, they prove the Burkholder martingale inequalities with the conditioned square function (of which the Rosenthal inequality is a special case) in full generality. Thirdly, they extend the dual form of Doob’s inequality from Junge’s paper (by reversing it) to the values \(0<p<1\). Finally, the non-faithful case and the order of growth of the involved constants are discussed.
Reviewer: Stanisław Goldstein (Łódź)
MSC:
| 46L53 | Noncommutative probability and statistics |
Keywords:
noncommutative martingale inequalities; noncommutative \(L_p\) spaces; noncommutative Burkholder inequality; noncommutative Rosenthal inequalityReferences:
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