In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M= 0 , we show that the following two-sided inequality holds for all 1 ≤ p' ∞: [Figure not available: see fulltext.] Here γ([[M]]t) is the L ²-norm of the unique Gaussian measure on X having [[M]]t(x∗,y∗):=[⟨M,x∗⟩,⟨M,y∗⟩]t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of (⋆) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (⋆) holds for all 0 ' p' ∞, and that for purely discontinuous martingales the right-hand side of (⋆) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (⋆) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (⋆) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic L^p- and ø-estimates, a precise construction of a decoupled tangent martin-gale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued re-sults in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapien, McConnell, and Woyczynski. The vast majority of the assertions presented in the paper is done under the necessary and sufficient UMD assumption on the corresponding Banach space.

In this paper we consider local martingales with values in a UMD Banach function space. We prove that such martingales have a version which is a martingale field. Moreover, a new Burkholder–Davis–Gundy type inequality is obtained.

In this paper we show that Musielak–Orlicz spaces are UMD spaces under the so-called Δ₂ condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak–Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that reflexive variable Lebesgue spaces L^p(·) are UMD spaces.

Let X be a given Banach space, and let M and N be two orthogonal X-valued local martingales such that N is weakly differentially subordinate to M. The paper contains the proof of the estimate E Ψ (Nt) ≤ CΦ, Ψ, X E Φ (Mt), t ≥ 0, where Φ, Ψ: X → R+ are convex continuous functions and the least admissible constant CΦ, Ψ, X coincides with the Φ, Ψ -norm of the periodic Hilbert transform. As a corollary, it is shown that the Φ, Ψ -norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if Φ is symmetric. We also prove that under certain natural assumptions on Φ and Ψ, the condition CΦ, Ψ, X < ∞ yields the UMD property of the space X. As an application, we provide comparison of Lp-norms of the periodic Hilbert transform to Wiener and Paley-Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

In this paper, we consider Meyer–Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞), any X-valued L^p-martingale M has a unique decomposition M = M^d + M^c such that M^d is a purely discontinuous martingale, M^c is a continuous martingale, M₀ ^c = 0 and EM_∞ ^{d p} + EM_∞ ^{c p} ≤ c_p,XEM_∞ ^p. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application, we show that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞) and for all X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate EN_∞ ^p ≤ C_p,XEM_∞ ^p

We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general X-valued local martingale is possible if and only if X has the UMD property. More precisely, X is a UMD Banach space if and only if for any X-valued local martingale M there exist a continuous local martingale M^c, a purely discontinuous quasi-left continuous local martingale M^q, and a purely discontinuous local martingale M^a with accessible jumps such that M = M^c + M^q + M^a. The corresponding weak L¹-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak L¹-bounds for a certain class of vector-valued continuous-time martingale transforms.

We introduce the notion of weak differential subordination for martingales, and show that a Banach space X is UMD if and only if for all p ∈ (1, ∞) and all purely discontinuous X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate E || N_∞ ||^p ≤ C_pE|| M_∞ ||_p. As a corollary we derive a sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.

In this paper, we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular, we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale (Formula presented.) there exists a time change (Formula presented.) such that (Formula presented.) is Brownian representable.