Perturbation of Burkholder’s martingale transform and Monge–Ampère equation

Given a sequence of martingale differences, Burkholder found the sharp constant for the LpLp-norm of the corresponding martingale transform. We are able to determine the sharp LpLp-norm of a small “quadratic perturbation” of the martingale transform in LpLp. By “quadratic perturbation” of the martingale transform, we mean the LpLp norm of the square root of the squares of the martingale transform and the original martingale (with a 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). Let {dk}k≥0{dk}k≥0 be a complex martingale difference in Lp[0,1]Lp[0,1], where 1