A Conjecture on Martingales and Rotations

We conjecture that continuous complex-valued piecewise affine maps of compact support in the complex plane have a probabilistic structure: associated with certain combinations of the first partial derivatives of such functions, there are two fields of rotations, and two martingales that are martingale transforms of each other, starting from constants of equal modulus, and ending at what one obtains after rotating these combinations of the derivatives. We prove this result in certain cases of continuous piecewise affine functions in the plane depending on 13 complex parameters. The motivation for this is that such a result would be sufficient to prove the conjectured value for the sharp p−norm of the Beurling–Ahlfors transformation in the plane. Indeed the result for the norms of these transformations would then follow from Burkholder’s estimates for the norms of two martingales that are martingale transforms of each other. On the other hand, it is shown that if we look for a way of obtaining the desired estimate for the norm of the Beurling–Ahlfors transformation, then we are naturally lead to considering martingales that are obtained after rotations from a function and its Beurling–Ahlfors transformation.