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

作者：Hinz Michael; Teplyaev Alexander^*

来源：Mathematische Nachrichten, 2018, 291(11-12): 1743-1756.

DOI：10.1002/mana.201600467

• Dirichlet metric measure spaces
• exterior derivative
• martingale dimension
• non-closable curl
• non self-similar Sierpinski carpets of Mackay
• Tyson and Wildrick

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.

• 出版日期2018-8