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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Maki, John M.

Contributors dc:contributor

• Tyson, Jeremy T.
• Wu, Jang-Mei
• D'Angelo, John P.
• Merenkov, Sergiy A.

• Heisenberg group
• Poincare domain
• s-John domain
• weak s-John domain
• Carnot groups
• Holder graphs
• Sobolev graphs

Statement dc:rights

• Copyright 2010 John Maki

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/26279

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/26279