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

Abstract

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 Lp-norm of a small “quadratic perturbation” of the martingale transform in Lp. By “quadratic perturbation” of the martingale transform, we mean the Lp 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 be a complex martingale difference in Lp[0,1], where 1<p<∞, and {εk}k≥0 a sequence in {±1}. We obtain the following generalization of Burkholder’s famous result. If τ∈[−12,12] and n∈Z+, then ‖∑k=0n(εkτ)dk‖Lp([0,1],C2)≤((p∗−1)2+τ2)12‖∑k=0ndk‖Lp([0,1],C), where ((p∗−1)2+τ2)12 is sharp and p∗−1=max{p−1,1p−1}. For 2≤p<∞, the result is also true with the sharp constant for τ∈R.