Dissertation/Thesis Abstract

Analysis in the Heisenberg group: Weak s-John domains and the dimensions of graphs of Hölder functions
by Maki, John Michael, Ph.D., University of Illinois at Urbana-Champaign, 2011, 128; 3496795
Abstract (Summary)

In this thesis, we provide connections between analytic properties in Euclidean [special characters omitted] 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 C1,α domain in [special characters omitted] 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 α. We follow upon this by showing that a weak s-John domain in a general Carnot group will be a (q, p)-Poincaré 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 Hölder function, with application to the dimension of a Sobolev graph.

Indexing (document details)
Advisor: Wu, Jang-Mei G.
Commitee:
School: University of Illinois at Urbana-Champaign
School Location: United States -- Illinois
Source: DAI-B 73/05, Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Carnot groups, Heisenberg groups, Holder graphs, Poincare domains
Publication Number: 3496795
ISBN: 9781267164650
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