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The Poincaré-Hardy Inequality on the Complement of a Cantor Set

  • Cristian S. Calude
  • Boris Pavlov
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

Abstract

The Poincare-Hardy inequality on the complement of the Cantor set E
$$\int {\frac{{\left| u \right|^2 }}{{dist^2 \left( {x,E} \right)}}dm \leqslant 4K^2 \cdot \int {\left| {\nabla u} \right|^2 dm} } $$
holds for every u 2 1 (R 3). Corresponding higher-order inequalities will be also derived. We use a special self-similar tiling and a natural metric on the space of trajectories generated by a Mauldin-Williams graph which is homeomorphic with the space of tiles endowed with the Euclidean distance. A crude estimation of the constant K is 2,100. Three applications will be briefly discussed. In the second one, the constant 1/2K -1 ≈ 0.0002 plays the role of an estimate for the dimensionless Planck constant in the corresponding uncertainty principle.

Keywords

Uncertainty Principle Binary String Quasiregular Mapping Minkowski Dimension Special Tiling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Cristian S. Calude
    • 1
  • Boris Pavlov
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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