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Alberti's type rank one theorem for martingales

Rami Ayoush, Dmitriy Stolyarov, Michał Wojciechowski

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.

Alberti's type rank one theorem for martingales

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 -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.
Paper Structure (3 sections, 5 theorems, 47 equations)

This paper contains 3 sections, 5 theorems, 47 equations.

Table of Contents

  1. Decomposition into flat and convex atoms
  2. Proof of the main theorem
  3. Martingale rank-one theorem

Key Result

Theorem 1

Let be a constant-coefficient linear operator that maps $\mathbb{R}^m$-valued functions in $N$ variables to $\mathbb{R}^n$-valued functions, with the principal symbol Suppose that a locally finite vector measure $\mu \in \mathop{\mathrm{M}}\nolimits(\Omega; \mathbb{R}^{m})$, where $\Omega \subset \mathbb{R}^{N}$ is an arbitrary domain, satisfies Then,

Theorems & Definitions (16)

  • Theorem 1: DR, Theorem 1.1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 10
  • ...and 6 more