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Weak type inequality for noncommutative differentially subordinated martingales

Abstract

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved.

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Author information

Correspondence to Adam Osȩkowski.

Additional information

Research supported by MEN Grant 1 PO3A 012 29.

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Osȩkowski, A. Weak type inequality for noncommutative differentially subordinated martingales. Probab. Theory Relat. Fields 140, 553–568 (2008). https://doi.org/10.1007/s00440-007-0075-0

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Keywords

  • Noncommutative probability space
  • Martingale
  • Weak type (1,1) inequality
  • Differentially subordinated martingales

Mathematics Subject Classification (2000)

  • Primary: 46L53
  • Secondary: 60G42