Orthogonality in complex martingale spaces and connections with the Beurling–Ahlfors transform

Abstract

We introduce and analyze a notion of orthogonality and dimension for spaces of $\mathbb{C}^n$-martingales. In particular, the space of martingale transforms of heat-extensions of $L^2(\mathbb{R}^{2m})$ functions is shown to be the orthogonal sum of $2$ conformal subspaces. We show that a theorem and proof of D. L. Burkholder for the computation of $L^p$-norm of martingale transforms applies specially for $n$-conformal and for pairwise conformal $n$-martingales. This leads to estimates of the $L^p$-norms of singular integral operators associated with the second-order Riesz transforms, in particular of the Beurling–Ahlfors operator.