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arXiv · 2107.05098

An elementary approach to the homological properties of constant-rank operators

Adolfo Arroyo-Rabasa·José Simental
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Abstract

We give a simple and constructive extension of Rai\c{t}\u{a}'s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincar\'e lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.

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Adolfo Arroyo-Rabasa, José Simental. 2021-07-11. An elementary approach to the homological properties of constant-rank operators. https://arxiv.org/abs/2107.05098

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