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

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

Maki, John M.

关键词: Heisenberg group;    Poincare domain;    s-John domain;    weak s-John domain;    Carnot groups;    Holder graphs;    Sobolev graphs;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/26279/Maki_John.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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-Johnis 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 resultis 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.