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

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Title: An elementary approach to the homological properties of constant-rank operators.
Authors: Arroyo-Rabasa, Adolfo1 adolfo.arroyo@uclouvain.be, Simental, José2 simental@im.unam.mx
Source: Comptes Rendus. Mathématique. 2023, Vol. 361, p45-63. 19p.
Subjects: Elliptic operators, Homogeneous spaces, Set-valued maps, Toeplitz operators
Abstract: We give a simple and constructive extension of Raiță'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 over the order of the operators constructed by Raiță and 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 discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over ℝd, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:We give a simple and constructive extension of Raiță'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 over the order of the operators constructed by Raiță and 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 discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over ℝd, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property. [ABSTRACT FROM AUTHOR]
ISSN:1631073X
DOI:10.5802/crmath.388