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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Harmonic calculus on fractals--A measure geometric approach II


Author: M. Zähle
Journal: Trans. Amer. Math. Soc. 357 (2005), 3407-3423
MSC (2000): Primary 28A80; Secondary 42B35, 47G30, 35P20
Published electronically: April 27, 2005
MathSciNet review: 2146630
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Abstract: Riesz potentials of fractal measures $\mu$ in metric spaces and their inverses are introduced. They define self-adjoint operators in the Hilbert space $L_2(\mu)$ and the former are shown to be compact.

In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.


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Additional Information

M. Zähle
Affiliation: Mathematical Institute, University of Jena, 07737 Jena, Germany
Email: zaehle@math.uni-jena.de

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03854-7
PII: S 0002-9947(05)03854-7
Keywords: Fractal set and measure, potential, pseudodifferential operator, Besov space, spectral asymptotics
Received by editor(s): October 4, 2000
Published electronically: April 27, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.