Esmayli, Behnam
(2021)
Geometric Function Theory in Metric Spaces.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
First, we generalize the coarea inequality, also known as Eilenberg's inequality, and provide a self-contained proof of it. The only previously known proof is based on a difficult result of Davies, which our proof avoids. Next, we find several equivalent conditions for Lipschitz functions from Euclidean cubes into arbitrary metric spaces to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul \cite{DS}. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if $f$ is a Lipschitz mapping from an open set in $\mathbb{R}^n$ onto a metric space $X$, then the topological dimension of $X$ equals $n$ if and only if $X$ has positive $n$-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, as a concrete application, a new formula for integration on countably rectifiable sets in the Heisenberg groups.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
8 October 2021 |
| Date Type: |
Publication |
| Defense Date: |
30 June 2021 |
| Approval Date: |
8 October 2021 |
| Submission Date: |
28 June 2021 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
111 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Hausdorff measure, weighted integrals, coarea inequality, metric derivative, area formula, coarea formula, mapping content, length preserving maps, Heisenberg groups, topological dimension, metric trees, factorization, quasiconvex metric spaces |
| Date Deposited: |
08 Oct 2021 19:32 |
| Last Modified: |
08 Oct 2021 19:32 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/41368 |
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