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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
Doctoral Committee Member(s):Tyson, Jeremy T.; D'Angelo, John P.; Merenkov, Sergiy A.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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
Rights Information:Copyright 2010 John Maki
Date Available in IDEALS:2013-08-27
Date Deposited:2011-08

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