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Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals


Authors: Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 361 (2009), 1765-1790
MSC (2000): Primary 28A80; Secondary 31C45, 60J60
DOI: https://doi.org/10.1090/S0002-9947-08-04772-7
Published electronically: November 24, 2008
MathSciNet review: 2465816
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Abstract: We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals; however the cutoff technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.


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Additional Information

Luke G. Rogers
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: rogers@math.uconn.edu

Robert S. Strichartz
Affiliation: Department of Mathematics, Cornell University, Malott Hall, Ithaca, New York 14853-4201
Email: str@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/S0002-9947-08-04772-7
Received by editor(s): January 4, 2007
Published electronically: November 24, 2008
Additional Notes: This research was supported in part by the National Science Foundation, Grant DMS-0140194
This research was supported in part by the National Science Foundation, Grant DMS-0505622
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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