Date of Completion

8-4-2020

Embargo Period

8-3-2020

Keywords

rectifiability, pointwise doubling measures, Hilbert space, metric space analysis, Traveling Salesman Theorem

Major Advisor

Matthew Badger

Associate Advisor

Vasileios Chousionis

Associate Advisor

Lan-Hsuan Huang

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In geometric measure theory, there is interest in understanding the interactions of measures with rectifiable sets. Here, we study such interactions in three settings. First, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure, we construct a multiresolution family of windows, and then we use a weighted Jones’ function to record how well lines approximate the distribution of mass in each window. We show that when the measure is rectifiable, the mass is sufficiently concentrated around a line at each scale and that the converse also holds. We relay an algorithm for the construction of a rectifiable curve through appropriately chosen nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many points that are separated by a fixed distance even in a bounded set. Second, we prove a characterization for pointwise doubling measures which are carried by Lipschitz graphs. In this characterization, we show that if the mass in small neighborhoods of each typical point is sufficiently concentrated within a cone then the measure is carried by Lipschitz graphs, and, again, the converse holds as well. Finally, we discuss a notion of fractional rectifiability in which rectifiable curves are replaced by images of Holder maps. We present a sufficient condition under which a pointwise doubling measure in infinite dimensional Hilbert space is carried by these Holder images.

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