Sharp L2 log L inequalities for the Haar system and martingale transforms

Let (h(n))(n >= 0) be the Haar system of functions on [0, 1]. The paper contains the proof of the estimate integral(1)(0) vertical bar Sigma(n)(k=0) epsilon(k)alpha(k)h(k)vertical bar(2) log vertical bar Sigma(n)(k=0) epsilon(k)alpha(k)h(k vertical bar) ds <= integral(1)(0) vertical bar Sigma(n)(k=0) alpha(k)h(k)vertical bar(2) log vertical bar e(2) Sigma(n)(k=0) alpha(k)h(k) vertical bar ds, for n = 0, 1, 2,.... Here (a(n))(n >= 0) is an arbitrary sequence with values in a given Hilbert space H and (epsilon(n))(n >= 0) is a sequence of signs. The constant e(2) appearing on the right is shown to be the best possible. This result is generalized to the sharp inequality E vertical bar g(n)vertical bar(2) log vertical bar g(n)vertical bar <= E vertical bar f(n)vertical bar(2) log(e(2)vertical bar fn vertical bar), n = 0, 1, 2,..., where (f(n))(n >= 0) is an arbitrary martingale with values in H and (g(n))(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 vertical bar f(n)vertical bar(2) log(e(-2) vertical bar f(n)vertical bar) <= ESn2 (f) log S-n(f) E vertical bar f(n)vertical bar(2) log(e(2)vertical bar f(n)vertical bar), n = 0, 1, 2,.... (C) 2014 Elsevier B.V. All rights reserved.