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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]
Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 175255418
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  Data: An elementary approach to the homological properties of constant-rank operators.
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  Data: <searchLink fieldCode="JN" term="%22Comptes+Rendus%2E+Mathématique%22">Comptes Rendus. Mathématique</searchLink>. 2023, Vol. 361, p45-63. 19p.
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  Data: <searchLink fieldCode="DE" term="%22Elliptic+operators%22">Elliptic operators</searchLink><br /><searchLink fieldCode="DE" term="%22Homogeneous+spaces%22">Homogeneous spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Set-valued+maps%22">Set-valued maps</searchLink><br /><searchLink fieldCode="DE" term="%22Toeplitz+operators%22">Toeplitz operators</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: 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]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.5802/crmath.388
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 19
        StartPage: 45
    Subjects:
      – SubjectFull: Elliptic operators
        Type: general
      – SubjectFull: Homogeneous spaces
        Type: general
      – SubjectFull: Set-valued maps
        Type: general
      – SubjectFull: Toeplitz operators
        Type: general
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      – TitleFull: An elementary approach to the homological properties of constant-rank operators.
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            NameFull: Arroyo-Rabasa, Adolfo
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            NameFull: Simental, José
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            – D: 01
              M: 01
              Text: 2023
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              Y: 2023
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            – Type: volume
              Value: 361
          Titles:
            – TitleFull: Comptes Rendus. Mathématique
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