Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

Luigi De Masi; Carlo Gasparetto

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arXiv · 2312.06026

Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

Luigi De Masi·Carlo Gasparetto

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Abstract

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $\mu$ satisfying $\mathcal{A}\mu=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $\mu$. Finally, we show that operators in the above class satisfy a Lusin-type property.

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Luigi De Masi, Carlo Gasparetto. 2023-12-10. Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures. https://arxiv.org/abs/2312.06026

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