Let (hn)n≥0 be the Haar system of functions on [0,1]. The paper contains the proof of the estimate ∫01|∑k=0nεkakhk|2log|∑k=0nεkakhk|ds≤∫01|∑k=0nakhk|2log|e2∑k=0nakhk|ds, for n=0,1,2,…. Here (an)n≥0 is an arbitrary sequence with values in a given Hilbert space H and...

Let f, g be two Hilbert-space-valued martingales such that g is differentially subordinate to f. The paper contains the proof of the estimate ‖g‖p,∞≤2p(p+1)p−1‖f‖p,∞,1p∞. The constant is shown to be of optimal order for p→∞ and for p→1. Related results for transforms of...

Let X be a continuous-path uniformly integrable martingale such that its exponential process E(X) satisfies the probabilistic version of Muckenhoupt’s condition A1. We establish optimal upper bounds for the BMO norm of X and a class of related sharp exponential estimates.

We determine the optimal constants in the weak type (p,q) inequalities involving a martingale, its square and conditional square function. Some applications are presented.

We study sharp square function inequalities for martingales taking values in a 2-convex Banach space B. We show that an appropriate weak-type bound holds true if and only if B is isometric to a Hilbert space.

Suppose that f is a martingale taking values in a Banach space B and g is its transform by a deterministic sequence of numbers in {−1,1}, such that supn‖gn‖≥1 almost surely. We show that a certain family of Φ-estimates for f holds true if and only B is a Hilbert space.

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 ‖supt≥0|Yt|‖1≤3‖supt≥0Xt‖1 and show that the constant 3 is the best possible.