Sharp LP-bounds for a perturbation of Burkholder's Martingale Transform

Let {d(k)}(k >= 1) be a real martingale difference in L-p[0, 1), where 1 < p < infinity, and {epsilon(k)}(k >= 1) subset of (+/- 1). We obtain the following generalization of Burkholder's famous result. If tau is an element of [-1/2, 1/2] and n is an element of Z(+) then parallel to Sigma(n)(k=1)(epsilon(k)d(k), tau d(k))parallel to(Lp((0,1),R2)) <= ((p* - 1)(2) + iota(2))(1/2) parallel to Sigma(n)(k=1)dk parallel to(Lp((0,1),R)) where ((p* - 1)(2) + tau(2))(1/2) is sharp and p* - 1 = max{p - 1, 1/p - 1}. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.