Sharp L2logL inequalities for the Haar system and martingale transforms

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 (εn)n≥0 is a sequence of signs. The constant e2 appearing on the right is shown to be the best possible. This result is generalized to the sharp inequality E|gn|2log|gn|≤E|fn|2log(e2|fn|),n=0,1,2,…, where (fn)n≥0 is an arbitrary martingale with values in H and (gn)n≥0 is its transform by a predictable sequence with values in {−1,1}. As an application, we obtain the two-sided bound for the martingale square function S(f): E|fn|2log(e−2|fn|)≤ESn2(f)logSn(f)≤E|fn|2log(e2|fn|),n=0,1,2,….