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Given two martingales on the filtration generated by two dimensional Brownianmotion, we want to estimate the $L^p$ norm of the subordinated one if we havesome extra orthogonality property available. We construct several new Bellmanfunctions, very different from Burkholder's function, and using them give anestimate of $L^p$ norm of a subordinated martingale, if the dominatingmartingale is orthogonal and $1<p\le 2$. We use Monge--Ampere equation toconstruct these Bellman functions.

Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1