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.
|Advisor:||Wu, Jang-Mei G.|
|School:||University of Illinois at Urbana-Champaign|
|School Location:||United States -- Illinois|
|Source:||DAI-B 73/05, Dissertation Abstracts International|
|Keywords:||Carnot groups, Heisenberg groups, Holder graphs, Poincare domains|
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