Burkholder obtained a sharp estimate of $\E|W|^p$ via $\E|Z|^p$, where $W$ is
a martingale transform of $Z$, or, in other words, for martingales $W$
differentially subordinated to martingales $Z$. His result is that $\E|W|^p\le
(p^*-1)^p\E|Z|^p$, where $p^* =\max (p, \frac{p}{p-1})$. What happens if the
martingales have an extra property of being orthogonal martingales? This
property is an analog (for martingales) of the Cauchy-Riemann equation for
functions, and it naturally appears from a problem on singular integrals (see
the references at the end of Section~1).