Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions
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.
Heisenberg group, Poincare domain, s-John domain, weak s-John domain, Carnot groups, Holder graphs, Sobolev graphs
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign