Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

Prabhu Janakiraman; Vasily Vasyunin; Alexander Volberg

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arXiv · 1012.0947

Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

Prabhu Janakiraman·Vasily Vasyunin·Alexander Volberg

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Abstract

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

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Prabhu Janakiraman, Vasily Vasyunin, Alexander Volberg. 2010-12-04. Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1<p\le 2$. https://arxiv.org/abs/1012.0947

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