Asymmetric Doob inequalities in continuous time

Abstract

Abstract. The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let (M, τ ) be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras (Mt) 0≤t≤1 whose union is weak- * dense in M. Let Et denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for 1 < p < 2 and x ∈ Lp(M, τ ) one can find a, b ∈ Lp(M, τ ) and contractions ut, vt ∈ M such that Et(x) = aut + vtb and max a p , b p ≤ cp x p. Moreover, aut and vtb converge in the row/column Hardy spaces H r p (M) and H c p (M) respectively. We also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations, including p = 1. As for other results in noncommutative martingale theory, the passage from discrete to continuous index is quite technical and requires genuinely new methods. Our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of Lp-modules. The convergence in H r p (M) and H c p (M) also imposes new algebraic atomic decompositions.

Noncommutative Good-$λ$ Inequalities

“…In particular, the noncommutative analogue of Gundy's decomposition of a martingale was obtained by Parcet and Randrianantoanina in [71] and a version of Davis' decomposition was found by Perrin in [72]; these have been greatly improved in very recent papers [82,84]. We also refer the reader to the important works on the weak-type versions of the estimates above, given by Randrianantoanina [79,80,81], certain noncommutative atomic decompositions [12] and its recent improvement together with a John-Nirenberg inequality by Hong and Mei [40], and some recent advances regarding algebra atomic decompositions and asymmetric Doob's inequalities by Junge et al [38,39]. Finally, we mention the works [9,10,11,42,83] for martingale inequalities in the context of various noncommutative symmetric spaces, the articles [44,45,46] for the noncommutative analogs of Johnson-Schechtman inequalities, and the very recent paper [47] for the duality of noncommutative dyadic martingale Hardy space.…”

Noncommutative pointwise convergence of orthogonal expansions of several variables

“…Chen-Xu-Yin [4] exploited Mei's result to prove maximal inequalities associated to the integrable rapidly decreasing functions. We refer to [12][13][14][15]20] for more information on the development of noncommutative harmonic analysis. We also refer to [3,11,[16][17][18]23]) for more information on related maximal inequalities.…”