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...

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.