Distributional inequalities for noncommutative martingales

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.     

A remark on closed noncommutative subspaces

Given an abelian category with arbitrary products, arbitrary coproducts, and a generator, we show that the closed subspaces (in the sense of A. L. Rosenberg) are parameterized by a suitably defined poset of ideals in the generator. In particular, the collection of closed subspaces is itself a small poset.

     

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space ${\mathsf{h}}_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of ${\mathsf{h}}_p$ for all $0<p<1$, and provide a constructive proof of the atomic decomposition for $p=1$ which resolves a main problem on the subject left open for the last twelve years. We also study ${(p,\infty )}_c$-atoms, and show that every ${(p,2)}_c$-atom can be decomposed into a sum of ${(p,\infty )}_c$-atoms; consequently, for every $0<p\le 1$, the ${(p,q)}_c$-atoms lead to the same atomic space for all $2\le q\le \infty$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space ${\mathsf{h}}_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.     

On the maximal inequalities for martingales involving two functions

Let and be nonnegative convex functions, and let and be the right continuous derivatives of and respectively. In this paper, we prove the equivalence of the following three conditions: (i) (ii) and (iii) s_0,$">where and are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

A remark on vorticity maximal function

In this note the following inequality is proved. For any nonnegative measure μH⁻¹(R²), xR² and 0<r<1, there holds

(1)

where C is a positive constant. Using (1) an estimate for the vorticity maximal function similar to the estimate in Majda [A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J. 42 (1993) 921–939] is established without assuming the initial vorticity having compact support. From this a more simple proof of the Delort's celebrated theorem [J.M. Delort, Existence de mappes de fourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991) 553–586] is presented.     

Weak type inequality for noncommutative differentially subordinated martingales

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. Research supported by MEN Grant 1 PO3A 012 29.     

Sharp maximal inequality for nonnegative martingales

Let X be a nonnegative martingale, let H be a predictable process taking values in [−1,1] and let Y be an Itô integral of H with respect to X. We establish the bound and show that the constant 3 is the best possible.     

Convergence of weighted averages of noncommutative martingales

Let x = (x _n ) _n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w _n ) _n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x _n ) _n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ _n (x)) _n?1 of x converges b.a.u. to the same limit under some condition, where σ _n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x _n ) _n?1 converges in L _p ( $\mathcal{M}$ ) if and only if (σ _n (x)) _n?1 converges in L _p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L ₁( $\mathcal{M}$ ).     

Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales

We prove that atomic decomposition for the Hardy spaces h₁ and H₁ is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales hp and bmo form interpolation scales with respect to both complex and real interpolations.     

A remark on the centered n-dimensional Hardy-Littlewood maximal function

We study the behaviour of the n-dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants c _n that appear in the weak type (1,1) inequalities.     

A sharp maximal inequality for continuous martingales and their differential subordinates

Assume that X, Y are continuous-path martingales taking values in ? ^ν , ν ? 1, such that Y is differentially subordinate to X. The paper contains the proof of the maximal inequality $$\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {Y_t } \right|} \right\|_1 \leqslant 2\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {X_t } \right|} \right\|_1 .$$ The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.     

A remark on convex functions andp-harmonic maps

We prove that the constant maps are the onlyp-harmonic maps for anyp 2 from an arbitrary compact Riemannian manifold into a complete Riemannian manifold which admits a strictly convex function.     

A remark on the approximation of plurisubharmonic functions

We show by an example that the Demailly approximation sequence of a plurisubharmonic function, constructed via Bergman kernels, is not a decreasing sequence in general.     

Random martingales and localization of maximal inequalities

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator