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Geometric Analysis on Metric Spaces

Malekzadeh, Soheil (2017) Geometric Analysis on Metric Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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My research is focused on analysis on metric spaces.
Over the past fifteen years, these subjects have expanded dramatically with applications far beyond pure mathematics in many branches of natural sciences. The generality of these ideas has reconciled a large number of previously singular attempts to extend tools from differential geometry to a much larger class of spaces that are not necessarily smooth.

This thesis includes three main results that have been the focus of my research at the University of Pittsburgh with the common theme of geometric analysis on metric spaces.

The Lusin's condition $(N)$ plays a significant role in the theory of integration.
Because of this importance in applications, we investigate how this condition is related to Sobolev spaces.
We focus our attention to the class of $W^{1,n}$ mappings and provide a new proof for the fact that continuous and pseudomonotone mapping of the class $W^{1,n}$ satisfies the condition $(N)$ on open sets. This result is due to Mal\'y and Martio but the original proof makes it difficult to gain insight into the internals of this result. We present a new proof which is based on the Hardy-Littlewood maximal function.

In the second result, we find necessary and sufficient conditions for a Lipschitz map $f:E\subset\mathbb{R}^n \to X$ into a metric space to satisfy $\mathcal{H}^k(f(E)) = 0.$
An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem.
Applications include pure unrectifiability of the Heisenberg groups and that of more general Carnot-Carath\'{e}odory spaces.

Lastly, we present a new characterization of the mappings of bounded length distortion (BLD for short).
In the original geometric definition it is assumed that a BLD mapping is open, discrete and sense preserving.
We prove that the first two of the three conditions are redundant and the sense-preserving condition can be replaced by a weaker assumption that the Jacobian is non-negative.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Malekzadeh, Soheilsom13@pitt.edusom13
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairHajlasz, Piotrhajlasz@pitt.eduhajlasz
Committee MemberDeBlois, Jasonjdeblois@pitt.edujdeblois
Committee MemberLennard, Christopherlennard@pitt.edulennard
Committee MemberTyson,
Date: 26 January 2017
Date Type: Publication
Defense Date: 15 November 2016
Approval Date: 26 January 2017
Submission Date: 29 November 2016
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 81
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: Analysis, Geometric analysis, Metric spaces, BLD mappings, Unrectifiability, Lusin condition N
Date Deposited: 26 Jan 2017 21:28
Last Modified: 27 Jan 2017 06:15

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