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Weak uncertainty principle for fractals, graphs and metric measure spaces


Authors: Kasso A. Okoudjou, Laurent Saloff-Coste and Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 360 (2008), 3857-3873
MSC (2000): Primary 28A80, 42C99; Secondary 26D99
DOI: https://doi.org/10.1090/S0002-9947-08-04472-3
Published electronically: February 27, 2008
MathSciNet review: 2386249
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Abstract: We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.


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Additional Information

Kasso A. Okoudjou
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Email: kasso@math.umd.edu

Laurent Saloff-Coste
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email: lsc@math.cornell.edu

Alexander Teplyaev
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: teplyaev@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04472-3
Keywords: Uncertainty principle, p.c.f. fractal, Heisenberg's inequality, measure metric spaces, Poincar{\'e} inequality, self-similar graphs, Sierpi{\'n}ski gasket, uniform finitely ramified graphs
Received by editor(s): August 15, 2006
Published electronically: February 27, 2008
Additional Notes: The second author was supported in part by NSF grant DMS-0603886
The third author was supported in part by NSF grant DMS-0505622
Article copyright: © Copyright 2008 American Mathematical Society

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