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

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Title: Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions
Author(s): Maki, John M.
Director of Research: Tyson, Jeremy T.
Doctoral Committee Chair(s): Wu, Jang-Mei G.
Doctoral Committee Member(s): Tyson, Jeremy T.; D'Angelo, John P.; Merenkov, Sergiy A.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(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.
Issue Date: 2011-08-26
URI: http://hdl.handle.net/2142/26279
Rights Information: Copyright 2010 John Maki
Date Available in IDEALS: 2013-08-27
Date Deposited: 2011-08
 

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