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.

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.

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.