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Geometric Function Theory in Metric Spaces

Esmayli, Behnam (2021) Geometric Function Theory in Metric Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Esmayli, Behnambee23@pitt.edubee23
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairHajlasz,
Committee MemberDeblois,
Committee MemberManfredi,
Committee MemberGoldstein,
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


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