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oai:www.ideals.illinois.edu:2142/26279 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Tyson, Jeremy T. Wu, Jang-Mei Tyson, Jeremy T. D'Angelo, John P. Merenkov, Sergiy A. Maki, John M. 2011-08-26T15:21:11Z 2013-08-27T10:00:19Z 2011-08 2011-08-26T15:21:11Z 2011-08 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. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2010-09-24T20:51:05Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Maki_John.pdf: 584172 bytes, checksum: 4b554684ea085c9077db53ffd765f582 (MD5) Made available in DSpace on 2011-08-26T15:21:11Z (GMT). No. of bitstreams: 2 Maki_John.pdf: 615287 bytes, checksum: edc14a8f8985b9f445c767095aeaf246 (MD5) license.txt: 4059 bytes, checksum: 98a0b00ae46990dc6fd60d5fe337568c (MD5) Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-08-26T15:25:36Z Item is restricted until 2013-08-26T15:25:28Z Item reinstated by Sarah Shreeves (sshreeve@illinois.edu) on 2013-08-27T10:00:19Z Item was in collections: University of Illinois Dissertations and Theses (ID: 204) Dissertations and Theses - Mathematics (ID: 749) No. of bitstreams: 3 Maki_John.pdf.txt: 173186 bytes, checksum: 3002048bcfdf9c9addada27c22c3e6da (MD5) Maki_John.pdf: 615287 bytes, checksum: edc14a8f8985b9f445c767095aeaf246 (MD5) license.txt: 4059 bytes, checksum: 98a0b00ae46990dc6fd60d5fe337568c (MD5) Item released from any restrictions by Sarah Shreeves (sshreeve@illinois.edu) on 2013-08-27T10:00:19Z http://hdl.handle.net/2142/26279 en Copyright 2010 John Maki Heisenberg group Poincare domain s-John domain weak s-John domain Carnot groups Holder graphs Sobolev graphs Analysis in the Heisenberg group: weak s-John domains and the dimensions of graphs of Holder functions Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD