Gundy's Decomposition for Non‐Commutative Martingales and Applications

By Javier Parcet, Narcisse Randrianantoanina. 2004

77 citations recorded by the source corpus.

We provide an analogue of Gundy's decomposition for L1‐bounded non‐commutative martingales. An important difference from the classical case is that for any L1‐bounded non‐commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non‐commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for non‐commutative martingale transforms and the non‐commutative analogue of Burkholder's weak type inequality for square functions. A sequence (xn)n⩾ 1 in a normed space X is called 2‐co‐lacunary if there exists a bounded linear map from the closed linear span of (xn)n ⩾ 1 to l2 taking each xn to the nth vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1 (M, τ) whose sequence of norms is bounded away from zero is 2‐co‐lacunary, generalizing a result of Aldous and Fremlin to non‐commutative L1‐spaces. 2000 Mathematics Subject Classification 46L53, 46L52 (primary), 46L51, 60G42 (secondary).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/272c3ebc1387ae53dfcec5f3918bdb5cc8ceec97

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Citations in the corpus

Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori (2015, background): “[17] was made possible by the recent developments on noncommutative martingale/ergodic inequalities and the Littlewood-Paley-Stein theory for quantum Markovian semigroups, which had been achieved thanks to the efforts of many researchers; see, for instance, [56, 31, 36, 37, 61, 62, 52], and [32, 44,”
Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory (2007, methodology): “We need a weak notion of support from [44] which is quite useful when dealing with weak type inequalities.”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “Our main tools to overcome it will be the noncommutative forms of Gundy’s and Calderón-Zygmund decompositions [24, 26] for martingales transforms and singular integral operators respectively.”
A reduction method for noncommutative L_p-spaces and applications (2008, methodology): “We refer, for instance, to [Ju1] for the Doob maximal inequality, to [JX1, JX2] for the Burkholder/Rosenthal inequalities, to [R1, R2, R3] for several weak type (1, 1) inequalities and to [PaR] for the Gundy decomposition.”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, background): “We mention only [JX2, PaR , R2, R3, R4], where, among many other results, the optimal orders of the best constants in most noncommutative martingale inequalities are determined.”
Conditioned square functions for noncommutative martingales (2005, methodology): “(i) In the proofs of Proposition A and Proposition B above, one can also use the non-commutative analogue of Gundy’s decomposition recently obtained in [37].”
Rosenthal Type inequalities for free chaos. (2005, methodology): “More precisely, according to the Khintchine and Rosenthal inequalities for free random variables, it is natural to ask whether or not the Burkholder-Gundy inequality [34] holds in the free setting for p = ∞, see also [17] for the non semifinite case and [28, 36] for the weak type (1, 1) inequality a”