University of Illinois Urbana-Champaign

Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions

Maki, John M.

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Maki_John.pdf

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https://hdl.handle.net/2142/26279

Description

Title
Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions
Author(s)
Maki, John M.
Issue Date
2011-08-26T15:21:11Z
Director of Research (if dissertation) or Advisor (if thesis)
Tyson, Jeremy T.
Doctoral Committee Chair(s)
Wu, Jang-Mei
Committee Member(s)
  • Tyson, Jeremy T.
  • D'Angelo, John P.
  • Merenkov, Sergiy A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2011-08-26T15:21:11Z
Keyword(s)
  • Heisenberg group
  • Poincare domain
  • s-John domain
  • weak s-John domain
  • Carnot groups
  • Holder graphs
  • Sobolev graphs
Abstract
In this thesis, we provide connections between analytic properties in Euclidean R^n and analytic properties in sub-Riemannian Carnot groups. We introduce weak s-John domains, in analogy with weak John domains, and we prove that weak s-John is equivalent to a localized version. This is applied in showing that a bounded C^{1,alpha} domain in R^3 will be a weak s-John domain in the first Heisenberg group. This result is sharp, giving a precise value of s that depends only on alpha. We follow upon this by showing that a weak s-John domain in a general Carnot group will be a (q,p)-Poincare domain for certain p and q that depend only on s and the homogeneous dimension of the Carnot group. The final result gives, in a general Carnot group, an upper bound on the lower box dimension of the graph of an Euclidean Holder function, with application to the dimension of a Sobolev graph.
Graduation Semester
2011-08
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http://hdl.handle.net/2142/26279
Copyright and License Information
Copyright 2010 John Maki

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