Given a sequence of martingale differences, Burkholder found the sharp constant for the Lp-norm of the corresponding martingale transform. We are able to determine the sharp L p-norm of small "quadratic perturbations" of the martingale transform in Lp. By "quadratic perturbation" of the martingale transform we mean the L p norm of the square root of the squares of the martingale transform and the original martingale (with small constant). The problem of perturbation of martingale transform appears naturally if one wants to estimate the linear combination of Riesz transforms (as, for example, in the case of Ahlfors-Beurling operator).
Read
