Doob's inequality for non-commutative martingales

By Marius Junge. 2002

294 citations recorded by the source corpus.

Let $1\le p<\8$ and $(x_n)_{\nen}$ be a sequence of positive elements in a non-commutative $L_p$ space and $(E_n)_{\nen}$ be an increasing sequence of conditional expectations, then the $L_p$ norm of \sum_n E_n(x_n) can be estimated by c_p times the $L_p$ norm of \sum_n x_n. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for $1

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba3f19332630ea54faccd3fd3f7c36e48aa209a3

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Citations in the corpus

Noncommutative Boyd interpolation theorems (2012, methodology): “3 (see the proof of [17], Proposition 3.”
Noncommutative maximal ergodic inequalities associated with doubling conditions (2017, background): “f the usual Bochner spaces Lp(X;ℓ∞). These vector-valued Lp-spaces were ﬁrst introduced by Pisier [Pis98] for injective von Neumann algebras and then extended to general von Neumann algebras by Junge [Jun02]. The descriptions and properties below can be found in [JX07, Section 2]. Given 1 ≤ p ≤ ∞, L”
Non-Commutative Martingale Transforms (2001, methodology): “Junge proved in [ 24 ] non-commutative versions of Doob’s maximal inequalities.”
Gundy's Decomposition for Non‐Commutative Martingales and Applications (2004, background): “Shortly after [24] was published, Junge obtained in [15] a non-commutative analogue of Doob’s maximal functions.”
A reduction method for noncommutative L_p-spaces and applications (2008, methodology): “1 Positive maps on Lp(M; l∞) We first recall the definition of the spaces Lp(M; l∞) and Lp(M; l1) from [Ju1].”
On ergodic theorems for free group actions on noncommutative spaces (2004, background): “The convergence of ( En(x) ) in the appropriate sense is a result due to Dang-Ngoc [10, Theorem 4] for p = +∞ and to Defant and Junge [11] in the two other cases.”
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, background): “A noncommutative analogue of Doob’s maximal function [24], the noncommutative John-Nirenberg theorem [26], extensions of Burkholder inequalities for conditioned square functions [30] and related weak type inequalities [50, 51, 52]; see [44] for a simpler approach to some of them.”
$H^{\infty }$ functional calculus and square functions on noncommutative $L^p$-spaces (2006, background): “See in particular [66, 33, 70, 37], and also the recent survey [81] and the references therein.”
A Weak Type Inequality for Non‐Commutative Martingales and Applications (2004, background): “A non-commutative analogue of Doob’s maximal inequality was successfully formulated and proved by Junge in [23] and non-commutative analogues of Burkholder/Rosenthal inequalities on conditioned square functions were studied by Junge and Xu in [25] among many other related topics.”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “We refer to [J1] for more information (see also [JX3]).”
BMO spaces associated with semigroups of operators (2011, background): “M) = ksup + t Et(|x −Etx|2)k 1 2 p 2 , BMO SPACES ASSOCIATED WITH SEMIGROUPS OF OPERATORS 15 where ksup+ ·kp 2 should be understood in the sense of vector-valued noncommutative Lpspaces, see [Pis98], [Jun02], [JX07]). Let kxkL pmo(M) = max{kxkLc pmo(M),kx ∗k Lc pmo(M)}. By Doob’s inequality, we know”
Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’ (2003, methodology): “on B. We denote by Dϕ◦E the density of ϕ◦ E. As for the three term K-functional, we deﬁne a new norm on L1(N) by kxkIK n(N,E) = infx=x 1+x2+x3 nkx1k1 + √ nkx2kLc 1(N,E) + √ nkx3kLr 1(N,E) . Following [J1], the space Lc 1(N,E) is deﬁned as the closure of NDϕ◦E with respect to the norm kzDϕ◦EkLc 1(N,E”
Rosenthal inequalities in noncommutative symmetric spaces (2011, methodology): “As a result we obtain a new proof of Rosenthal’s theorem for (Haagerup) Lp-spaces.”
Noncommutative Bennett and Rosenthal inequalities (2011, unknown): “NONCOMMUTATIVE BENNETT AND ROSENTHAL INEQUALITIES 5 According to [29] and [19], the norm of (xj) in Lp(l∞) is given by inf{‖a‖2p‖b‖2p} such that xj = ayjb with ‖yj‖∞ ≤ 1 .”