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arXiv · 2307.11381

Alberti's type rank one theorem for martingales

Rami Ayoush·Dmitriy Stolyarov·Michał Wojciechowski
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Abstract

We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the $q$-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti's rank-one theorem.

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Rami Ayoush, Dmitriy Stolyarov, Michał Wojciechowski. 2023-07-21. Alberti's type rank one theorem for martingales. https://doi.org/10.2140/pjm.2025.334.1

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