Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals
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- by Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev PDF
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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.References
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Additional Information
- Luke G. Rogers
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- MR Author ID: 785199
- 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
- MR Author ID: 361814
- Email: teplyaev@math.uconn.edu
- 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 - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2465816