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Densely defined non‐closable curl on carpet‐like metric measure spaces

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  • Michael Hinz
  • Alexander Teplyaev

Abstract

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.

Suggested Citation

  • Michael Hinz & Alexander Teplyaev, 2018. "Densely defined non‐closable curl on carpet‐like metric measure spaces," Mathematische Nachrichten, Wiley Blackwell, vol. 291(11-12), pages 1743-1756, August.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:11-12:p:1743-1756
    DOI: 10.1002/mana.201600467
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