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Archiv der Mathematik

, Volume 79, Issue 1, pp 46–50 | Cite as

Calculus on linear Cantor sets

  • J. Schweizer
  • B. Frank
Article
  • 42 Downloads

Abstract.

We construct a calculus of 1-forms on Cantor subsets \( K \subseteqq \mathbb{R} \) of the real line. More precisely, for every \( p \in [1, \infty] \), we construct a Banach C(K)-bimodule \( \Omega^{p} (K) \), a closed derivation \( d: D(d) \subseteqq C(K) \rightarrow \Omega^{p}(K) \), and continuous functionals \( \int\limits_{x}^{y}: \Omega^{p}(K) \rightarrow \mathbb{C} \) such that \( \int\limits_{x}^{y} df = f(y) - f(x), x, y \in K, f \in D(d) \). For \( p = 2, \Omega^{2}(K) \) is a Hilbert space, and choosing a Borel measure μ with suppμ = K, \( d: D(d) \subseteqq L^{2}(K) \rightarrow \Omega^{2}(K) \) is adjointable and \( \Delta_{\mu} = d^{*}d \) coincides with a diffusion operator introduced by Feller.

Keywords

Hilbert Space Real Line Borel Measure Diffusion Operator Continuous Functional 

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Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • J. Schweizer
    • 1
  • B. Frank
    • 1
  1. 1.Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany,¶ e-mail: juergen.schweizer@uni-tuebingen.deGermany

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