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A Lebesgue–Lusin property for linear operators of first and second order

Published online by Cambridge University Press:  06 November 2023

Adolfo Arroyo-Rabasa*
Université catholique de Louvain, Louvain La Neuve, Belgium (


We prove that for a homogeneous linear partial differential operator $\mathcal {A}$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying

\[ \mathcal{A} u(x)= f(x) \qquad \text{almost everywhere}. \]
This extends a result of G. Alberti for gradients on $\mathbf {R}^N$. In particular, for $0 \le m < N$, it is shown that every integrable $m$-vector field is the absolutely continuous part of the boundary of a normal $(m+1)$-current.

Research Article
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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