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Preservation of bounded geometry under transformations metric spaces

Li, Xining
http://orcid.org/0000-0001-6956-3926
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722

Abstract Details

2015, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
In the theory of geometric analysis on metric measure spaces, two properties of a metric measure space make the theory richer. These two properties are the doubling property of the measure, and the support of a Poincar&eacute inequality by the metric measure space. The focus of this dissertation is to show that the doubling property of the measure and the support of a Poincar&eacute inequality are preserved by two transformations of the metric measure space: sphericalization (to obtain a bounded space from an unbounded space), and flattening (to obtain an unbounded space from a bounded space). We will show that if the given metric measure space is equipped with an Ahlfors Q-regular measure, then so are the spaces obtained by the sphericalization/flattening transformations. We then show that even if the measure is not Ahlfors regular, if it is doubling, then the transformed measure is still doubling. We then show that if the given metric space satisfies an annular quaisconvexity property and the measure is doubling, and in addition if the metric measure space supports a p-Poincar&eacute inequality in the sense of Heinonen and Koskela's theory, then so does the transformed metric measure space (under the sphericalization/flattening procedure). Finally, we show that if we relax the annular quasiconvexity condition to an analog of the starlike condition for the metric measure space, then if the metric measure space also satisfies a p-Poincar&eacute inequality, the transformed space also must satisfy a q-Poincar&eacute inequality for some p≤ q< ∞. We also show that under a weaker version of the starlikeness hypothesis, support of ∞-Poincar&eacute inequality is preserved under the sphericalization/flattening procedure. We also provide some examples to show that the assumptions of annular quasiconvexity and the various versions of starlikeness conditions are needed in the respective results.
Nageswari Shanmugalingam, Ph.D. (Committee Chair)
Xiangdong Xie, Ph.D. (Committee Member)
Michael Goldberg, Ph.D. (Committee Member)
Andrew Lorent, Ph.D. (Committee Member)
Leonid Slavin, Ph.D. (Committee Member)
128 p.
Mathematics
Quasiconvexity and annular quasiconvexity; Upper gradient; Sphericalization and flattening; Poincare inequality; Radial starlike and meridean-like quasiconvex; Doubling measure

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Citations

  • Li, X. (2015). Preservation of bounded geometry under transformations metric spaces [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722

    APA Style (7th edition)

  • Li, Xining. Preservation of bounded geometry under transformations metric spaces. 2015. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722.

    MLA Style (8th edition)

  • Li, Xining. "Preservation of bounded geometry under transformations metric spaces." Doctoral dissertation, University of Cincinnati, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722

    Chicago Manual of Style (17th edition)

ucin1439309722
345
© 2015, some rights reserved.Creative Commons License Preservation of bounded geometry under transformations metric spaces by Xining Li is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Cincinnati and OhioLINK.