Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions

Maki, John M.

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https://hdl.handle.net/2142/26279 Copy

Description

Title

Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions

Author(s)

Maki, John M.

Issue Date

2011-08-26T15:21:11Z

Director of Research (if dissertation) or Advisor (if thesis)

Tyson, Jeremy T.

Doctoral Committee Chair(s)

Wu, Jang-Mei

Committee Member(s)

• Tyson, Jeremy T.
• D'Angelo, John P.
• Merenkov, Sergiy A.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2011-08

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

Copyright and License Information

Copyright 2010 John Maki

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