Subordination by Orthogonal Martingales

We are given two martingales on the filtration of the two dimensional Brownian motion. One is subordinated to another. We want to give an estimate of Lp-norm of a subordinated one via the same norm of a dominating one. In this setting this was done by Burkholder in [Bu1]–[Bu8]. If one of the martingales is orthogonal, the constant should drop. This was demonstrated in [BaJ1], when the orthogonality is attached to the subordinated martingale and when 2 ≤ p <∞. This note contains an (almost obvious) observation that the same idea can be used in the case when the orthogonality is attached to a dominating martingale and 1 < p ≤ 2. Two other complementary regimes are considered in [BJVLa]. When both martingales are orthogonal, see [BJVLe]. In these two papers the constants are sharp. We are not sure of the sharpness of the constant in the present note. A complex-valued martingale Y = Y1 + iY2 is said to be orthogonal if the quadratic variations of the coordinate martingales are equal and their mutual covariation is 0: 〈Y1〉 = 〈Y2〉 , 〈Y1, Y2〉 = 0. In [BaJ1], Bañuelos and Janakiraman make the observation that the martingale associated with the Beurling-Ahlfors transform is in fact an orthogonal martingale. They show that Burkholder’s proof in [Bu3] naturally accommodates for this property and leads to an improvement in the estimate of ‖B‖p.